Distributions 

Same as sample except that any intermediate computations are returned (useful for TransformedDistribution).

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

to_event(reinterpreted_batch_ndims: int | None = None) → Distribution[source]

Interpret the rightmost reinterpreted_batch_ndims batch dimensions as dependent event dimensions.

Parameters:: reinterpreted_batch_ndims – Number of rightmost batch dims to interpret as event dims.
Returns:: An instance of Independent distribution.
Return type:: numpyro.distributions.distribution.Independent

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

expand(batch_shape: tuple[int, ...]) → Distribution[source]

Returns a new ExpandedDistribution instance with batch dimensions expanded to batch_shape.

Parameters:: batch_shape (tuple) – batch shape to expand to.
Returns:: an instance of ExpandedDistribution.
Return type:: ExpandedDistribution

expand_by(sample_shape: tuple[int, ...]) → Distribution[source]

Expands a distribution by adding sample_shape to the left side of its batch_shape. To expand internal dims of self.batch_shape from 1 to something larger, use expand() instead.

Parameters:: sample_shape (tuple) – The size of the iid batch to be drawn from the distribution.
Returns:: An expanded version of this distribution.
Return type:: ExpandedDistribution

mask(mask: Array) → MaskedDistribution[source]

Masks a distribution by a boolean or boolean-valued array that is broadcastable to the distributions Distribution.batch_shape .

Parameters:: mask (bool or jnp.ndarray) – A boolean or boolean valued array (True includes a site, False excludes a site).
Returns:: A masked copy of this distribution.
Return type:: MaskedDistribution

Example:

>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.distributions import constraints
>>> from numpyro.infer import SVI, Trace_ELBO

>>> def model(data, m):
...     f = numpyro.sample("latent_fairness", dist.Beta(1, 1))
...     with numpyro.plate("N", data.shape[0]):
...         # only take into account the values selected by the mask
...         masked_dist = dist.Bernoulli(f).mask(m)
...         numpyro.sample("obs", masked_dist, obs=data)


>>> def guide(data, m):
...     alpha_q = numpyro.param("alpha_q", 5., constraint=constraints.positive)
...     beta_q = numpyro.param("beta_q", 5., constraint=constraints.positive)
...     numpyro.sample("latent_fairness", dist.Beta(alpha_q, beta_q))


>>> data = jnp.concatenate([jnp.ones(5), jnp.zeros(5)])
>>> # select values equal to one
>>> masked_array = jnp.where(data == 1, True, False)
>>> optimizer = numpyro.optim.Adam(step_size=0.05)
>>> svi = SVI(model, guide, optimizer, loss=Trace_ELBO())
>>> svi_result = svi.run(random.key(0), 300, data, masked_array)
>>> params = svi_result.params
>>> # inferred_mean is closer to 1
>>> inferred_mean = params["alpha_q"] / (params["alpha_q"] + params["beta_q"])

classmethod infer_shapes(*args: Any, **kwargs: Any) → tuple[tuple[int, ...], tuple[int, ...]][source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property is_discrete: bool

ExpandedDistribution

class ExpandedDistribution(base_dist: Distribution, batch_shape: tuple[int, ...] = ())[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {}

pytree_data_fields: tuple[str, ...] = ('base_dist',)

pytree_aux_fields: tuple[str, ...] = ('_expanded_sizes', '_interstitial_sizes', 'has_enumerate_support')

base_dist: Distribution

property has_rsample: bool

rsample(key: prng_key | None, sample_shape: tuple[int, ...] = ()) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Same as sample except that any intermediate computations are returned (useful for TransformedDistribution).

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

FoldedDistribution

class FoldedDistribution(base_dist: Distribution, *, validate_args: bool | None = None)[source]

Equivalent to TransformedDistribution(base_dist, AbsTransform()), but additionally supports log_prob() .

Parameters:: base_dist (Distribution) – A univariate distribution to reflect.

support = Positive(lower_bound=0.0)

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

ImproperUniform

class ImproperUniform(support: constraints.Constraint, batch_shape: tuple[int, ...], event_shape: tuple[int, ...], *, validate_args: bool | None = None)[source]

Bases: Distribution

A helper distribution with zero log_prob() over the support domain.

Note

sample method is not implemented for this distribution. In autoguide and mcmc, initial parameters for improper sites are derived from init_to_uniform or init_to_value strategies.

Usage:

>>> from numpyro import sample
>>> from numpyro.distributions import ImproperUniform, Normal, constraints
>>>
>>> def model():
...     # ordered vector with length 10
...     x = sample('x', ImproperUniform(constraints.ordered_vector, (), event_shape=(10,)))
...
...     # real matrix with shape (3, 4)
...     y = sample('y', ImproperUniform(constraints.real, (), event_shape=(3, 4)))
...
...     # a shape-(6, 8) batch of length-5 vectors greater than 3
...     z = sample('z', ImproperUniform(constraints.greater_than(3), (6, 8), event_shape=(5,)))

If you want to set improper prior over all values greater than a, where a is another random variable, you might use

>>> def model():
...     a = sample('a', Normal(0, 1))
...     x = sample('x', ImproperUniform(constraints.greater_than(a), (), event_shape=()))

or if you want to reparameterize it

>>> from numpyro.distributions import TransformedDistribution, transforms
>>> from numpyro.handlers import reparam
>>> from numpyro.infer.reparam import TransformReparam
>>>
>>> def model():
...     a = sample('a', Normal(0, 1))
...     with reparam(config={'x': TransformReparam()}):
...         x = sample('x',
...                    TransformedDistribution(ImproperUniform(constraints.positive, (), ()),
...                                            transforms.AffineTransform(a, 1)))

Parameters:

support (Constraint) – the support of this distribution.
batch_shape (tuple) – batch shape of this distribution. It is usually safe to set batch_shape=().
event_shape (tuple) – event shape of this distribution.

arg_constraints: dict[str, Any] = {}

pytree_data_fields: tuple[str, ...] = ('support',)

support: Constraint = Dependent()

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

Independent

class Independent(base_dist: Distribution, reinterpreted_batch_ndims: int, *, validate_args: bool | None = None)[source]

Bases: Distribution

Reinterprets batch dimensions of a distribution as event dims by shifting the batch-event dim boundary further to the left.

From a practical standpoint, this is useful when changing the result of log_prob(). For example, a univariate Normal distribution can be interpreted as a multivariate Normal with diagonal covariance:

>>> import numpyro.distributions as dist
>>> normal = dist.Normal(jnp.zeros(3), jnp.ones(3))
>>> [normal.batch_shape, normal.event_shape]
[(3,), ()]
>>> diag_normal = dist.Independent(normal, 1)
>>> [diag_normal.batch_shape, diag_normal.event_shape]
[(), (3,)]

Parameters:

base_distribution (numpyro.distribution.Distribution) – a distribution instance.
reinterpreted_batch_ndims (int) – the number of batch dims to reinterpret as event dims.

arg_constraints: dict[str, Any] = {}

pytree_data_fields: tuple[str, ...] = ('base_dist',)

pytree_aux_fields: tuple[str, ...] = ('reinterpreted_batch_ndims', 'has_enumerate_support')

base_dist: Distribution

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property has_rsample: bool

rsample(key: prng_key | None, sample_shape: tuple[int, ...] = ()) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

expand(batch_shape: tuple[int, ...]) → Distribution[source]

Returns a new ExpandedDistribution instance with batch dimensions expanded to batch_shape.

Parameters:: batch_shape (tuple) – batch shape to expand to.
Returns:: an instance of ExpandedDistribution.
Return type:: ExpandedDistribution

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

MaskedDistribution

class MaskedDistribution(base_dist: Distribution, mask: bool | Array)[source]

Bases: Distribution

Masks a distribution by a boolean array that is broadcastable to the distribution’s Distribution.batch_shape. In the special case mask is False, computation of log_prob() , is skipped, and constant zero values are returned instead.

Parameters:: mask (jnp.ndarray or bool) – A boolean or boolean-valued array.

arg_constraints: dict[str, Any] = {}

pytree_data_fields: tuple[str, ...] = ('base_dist', '_mask')

pytree_aux_fields: tuple[str, ...] = ('_mask', 'has_enumerate_support')

base_dist: Distribution

property has_rsample: bool

rsample(key: prng_key | None, sample_shape: tuple[int, ...] = ()) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

tree_flatten() → tuple[tuple[Any, ...], tuple[Any, ...]][source]

classmethod tree_unflatten(aux_data: tuple[Any, ...], params: tuple[Any, ...]) → MaskedDistribution[source]

TransformedDistribution

class TransformedDistribution(base_distribution: Distribution, transforms: Transform | list[Transform], *, validate_args: bool | None = None)[source]

Bases: Distribution

Returns a distribution instance obtained as a result of applying a sequence of transforms to a base distribution. For an example, see LogNormal and HalfNormal.

Parameters:

base_distribution – the base distribution over which to apply transforms.
transforms – a single transform or a list of transforms.
validate_args – Whether to enable validation of distribution parameters and arguments to .log_prob method.

arg_constraints: dict[str, Any] = {}

pytree_data_fields: tuple[str, ...] = ('base_dist', 'transforms')

transforms: list[Transform]

base_dist: Distribution

property has_rsample: bool

rsample(key: prng_key | None, sample_shape: tuple[int, ...] = ()) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Same as sample except that any intermediate computations are returned (useful for TransformedDistribution).

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

Delta

class Delta(v: ArrayLike = 0.0, log_density: ArrayLike = 0.0, event_dim: int = 0, *, validate_args: bool | None = None)[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {'log_density': Real(), 'v': Dependent()}

reparametrized_params: list[str] = ['v', 'log_density']

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Unit

class Unit(log_factor: ArrayLike, *, validate_args: bool = False)[source]

Bases: Distribution

Trivial nonnormalized distribution representing the unit type.

The unit type has a single value with no data, i.e. value.size == 0.

This is used for numpyro.factor() statements.

arg_constraints: dict[str, Any] = {'log_factor': Real()}

support = Real()

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

Continuous Distributions

AsymmetricLaplace

Bases: Distribution

arg_constraints: dict[str, Any] = {'asymmetry': Positive(lower_bound=0.0), 'loc': Real(), 'scale': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['loc', 'scale', 'asymmetry']

support = Real()

left_scale()[source]

right_scale()[source]

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

AsymmetricLaplaceQuantile

Bases: Distribution

An alternative parameterization of AsymmetricLaplace commonly applied in Bayesian quantile regression.

Instead of the asymmetry parameter employed by AsymmetricLaplace, to define the balance between left- versus right-hand sides of the distribution, this class utilizes a quantile parameter, which describes the proportion of probability density that falls to the left-hand side of the distribution.

The scale parameter is also interpreted slightly differently than in AsymmetricLaplace. When loc=0 and scale=1, AsymmetricLaplace(0,1,1) is equivalent to Laplace(0,1), while AsymmetricLaplaceQuantile(0,1,0.5) is equivalent to Laplace(0,2).

arg_constraints: dict[str, Any] = {'loc': Real(), 'quantile': OpenInterval(lower_bound=0.0, upper_bound=1.0), 'scale': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['loc', 'scale', 'quantile']

support = Real()

pytree_data_fields: tuple[str, ...] = ('loc', 'scale', 'quantile', '_ald')

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

Beta

Bases: Distribution

arg_constraints: dict[str, Any] = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['concentration1', 'concentration0']

support = UnitInterval(lower_bound=0.0, upper_bound=1.0)

pytree_data_fields: tuple[str, ...] = ('concentration0', 'concentration1', '_dirichlet')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

BetaProportion

Bases: Beta

The BetaProportion distribution is a reparameterization of the conventional Beta distribution in terms of a the variate mean and a precision parameter.

Reference:

Beta regression for modelling rates and proportion, Ferrari Silvia, and: Francisco Cribari-Neto. Journal of Applied Statistics 31.7 (2004): 799-815.

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'mean': OpenInterval(lower_bound=0.0, upper_bound=1.0)}

reparametrized_params: list[str] = ['mean', 'concentration']

support = UnitInterval(lower_bound=0.0, upper_bound=1.0)

pytree_data_fields: tuple[str, ...] = ('concentration',)

CAR

Bases: Distribution

The Conditional Autoregressive (CAR) distribution is a special case of the multivariate normal in which the precision matrix is structured according to the adjacency matrix of sites. The amount of autocorrelation between sites is controlled by correlation. The distribution is a popular prior for areal spatial data.

Parameters:

loc (float or ndarray) – mean of the multivariate normal
correlation (float) – autoregression parameter. For most cases, the value should lie between 0 (sites are independent, collapses to an iid multivariate normal) and 1 (perfect autocorrelation between sites), but the specification allows for negative correlations.
conditional_precision (float) – positive precision for the multivariate normal
adj_matrix (ndarray or scipy.sparse.csr_matrix) – symmetric adjacency matrix where 1 indicates adjacency between sites and 0 otherwise. jax.numpy.ndarray adj_matrix is supported but is not recommended over numpy.ndarray or scipy.sparse.spmatrix.
is_sparse (bool) – whether to use a sparse form of adj_matrix in calculations (must be True if adj_matrix is a scipy.sparse.spmatrix)

arg_constraints: dict[str, Any] = {'adj_matrix': Dependent(), 'conditional_precision': Positive(lower_bound=0.0), 'correlation': OpenInterval(lower_bound=-1, upper_bound=1), 'loc': RealVector(Real(), 1)}

support = RealVector(Real(), 1)

reparametrized_params: list[str] = ['loc', 'correlation', 'conditional_precision', 'adj_matrix']

pytree_aux_fields: tuple[str, ...] = ('is_sparse', 'adj_matrix')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

precision_matrix()[source]

static infer_shapes(loc, correlation, conditional_precision, adj_matrix)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

tree_flatten()[source]

classmethod tree_unflatten(aux_data, params)[source]

Cauchy

Bases: Distribution

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Real()

reparametrized_params: list[str] = ['loc', 'scale']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Chi2

Bases: Gamma

arg_constraints: dict[str, Any] = {'df': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['df']

CirculantNormal

Multivariate normal distribution with covariance matrix \(\mathbf{C}\) that is positive-definite and circulant [1], i.e., has periodic boundary conditions. The density of a sample \(\mathbf{x}\in\mathbb{R}^n\) is the standard multivariate normal density

\[p\left(\mathbf{x}\mid\boldsymbol{\mu},\mathbf{C}\right) = \frac{\left(\mathrm{det}\,\mathbf{C}\right)^{-1/2}}{\left(2\pi\right)^{n / 2}} \exp\left(-\frac{1}{2}\left(\mathbf{x}-\boldsymbol{\mu}\right)^\intercal \mathbf{C}^{-1}\left(\mathbf{x}-\boldsymbol{\mu}\right)\right),\]

where \(\mathrm{det}\) denotes the determinant and \(^\intercal\) the transpose. Circulant matrices can be diagnolized efficiently using the discrete Fourier transform [1], allowing the log likelihood to be evaluated in \(n \log n\) time for \(n\) observations [2].

Parameters:

loc – Mean of the distribution \(\boldsymbol{\mu}\).
covariance_row – First row of the circulant covariance matrix \(\boldsymbol{C}\). Because of periodic boundary conditions, the covariance matrix is fully determined by its first row (see jax.scipy.linalg.toeplitz() for further details).
covariance_rfft – Real part of the real fast Fourier transform of covariance_row, the first row of the circulant covariance matrix \(\boldsymbol{C}\).

References:

Wikipedia. (n.d.). Circulant matrix. Retrieved March 6, 2025, from https://en.wikipedia.org/wiki/Circulant_matrix
Wood, A. T. A., & Chan, G. (1994). Simulation of Stationary Gaussian Processes in \(\left[0, 1\right]^d\). Journal of Computational and Graphical Statistics, 3(4), 409–432. https://doi.org/10.1080/10618600.1994.10474655

arg_constraints: dict[str, Any] = {'covariance_rfft': IndependentConstraint(Positive(lower_bound=0.0), 1), 'covariance_row': PositiveDefiniteCirculantVector(), 'loc': RealVector(Real(), 1)}

support = RealVector(Real(), 1)

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

covariance_row() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

covariance_matrix() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

variance() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Variance of the distribution.

static infer_shapes(loc: tuple[int, ...] = (), covariance_row: tuple[int, ...] | None = None, covariance_rfft: tuple[int, ...] | None = None)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Dagum

Bases: Distribution

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'scale': Positive(lower_bound=0.0), 'sharpness': Positive(lower_bound=0.0)}

support = Positive(lower_bound=0.0)

reparametrized_params: list[str] = ['concentration', 'sharpness', 'scale']

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

sample(key: prng_key, sample_shape: tuple[int, ...] = ()) → Array[source]

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

Dirichlet

class Dirichlet(concentration: Array, *, validate_args: bool | None = None)[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {'concentration': IndependentConstraint(Positive(lower_bound=0.0), 1)}

reparametrized_params: list[str] = ['concentration']

support = Simplex()

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

static infer_shapes(concentration)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

[1] https://en.wikipedia.org/wiki/Euler-Maruyama_method

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

EulerMaruyama

class EulerMaruyama(t: Array, sde_fn: Callable[[Array, Array], tuple[Array, Array]], init_dist: Distribution, *, validate_args: bool | None = None)[source]

Bases: Distribution

Euler–Maruyama method is a method for the approximate numerical solution of a stochastic differential equation (SDE)

Parameters:

t (ndarray) – discretized time
sde_fn (callable) – function returning the drift and diffusion coefficients of SDE
init_dist (Distribution) – Distribution for initial values.

References

arg_constraints: dict[str, Any] = {'t': OrderedVector()}

pytree_data_fields: tuple[str, ...] = ('t', 'init_dist')

pytree_aux_fields: tuple[str, ...] = ('sde_fn',)

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

Exponential

Bases: Distribution

reparametrized_params: list[str] = ['rate']

arg_constraints: dict[str, Any] = {'rate': Positive(lower_bound=0.0)}

support = Positive(lower_bound=0.0)

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Gamma

Bases: Distribution

Implementation of the Gamma distribution, \(\mathrm{Gamma}(\alpha, \lambda)\), where, \(\alpha\) is the concentration and \(\lambda\) is the rate.

Parameters:

concentration (ArrayLike) – concentration parameter \(\alpha\) (also known as shape parameter).
rate (ArrayLike) – rate parameter \(\lambda\) (inverse scale parameter).

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}

support = Positive(lower_bound=0.0)

reparametrized_params: list[str] = ['concentration', 'rate']

sample(key: prng_key, sample_shape: tuple[int, ...] = ()) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Method to generate samples \(X \sim \mathrm{Gamma}(\alpha, \lambda)\). It uses gamma() under the hood to generate samples.

If \(X \sim \mathrm{Gamma}(\alpha, \lambda)\), then

\[f_X(x\mid \alpha, \lambda) = \frac{\lambda^{\alpha} x^{\alpha - 1} e^{-\lambda x}}{\Gamma(\alpha)}, \quad x > 0\]

It uses gammaln() to compute the logarithm of the gamma function.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: If \(X \sim \mathrm{Gamma}(\alpha, \lambda)\), then

\[\mathbb{E}[X] = \frac{\alpha}{\lambda}\]

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: If \(X \sim \mathrm{Gamma}(\alpha, \lambda)\), then

\[\mathrm{Var}[X] = \frac{\alpha}{\lambda^2}\]

cdf(x)[source]

If \(X \sim \mathrm{Gamma}(\alpha, \lambda)\), then

\[F_X(x \mid \alpha, \lambda) = \frac{1}{\Gamma(\alpha)} \gamma\left(\alpha, \lambda x\right)\]

where, \(\gamma(\cdot,\cdot)\) is the lower incomplete gamma function. This method uses regularized incomplete gamma function, which is implemented as gammainc().

If \(X \sim \mathrm{Gamma}(\alpha, \lambda)\), then

\[F^{-1}_X(q \mid \alpha, \lambda) = \frac{1}{\lambda} \gamma^{-1}\left(\alpha, q \Gamma(\alpha)\right)\]

where, \(\gamma^{-1}(\cdot,\cdot)\) is the inverse of the lower incomplete gamma function. This method uses regularized incomplete gamma inverse function, which is implemented as gammaincinv().

If \(X \sim \mathrm{Gamma}(\alpha, \lambda)\), then

\[H[X] = \alpha - \ln(\lambda) + \ln\Gamma(\alpha) + (1 - \alpha) \psi(\alpha)\]

where, \(\psi(\cdot)\) is the digamma function. This methods uses which is implemented as digamma().

GaussianCopula

class GaussianCopula(marginal_dist: Distribution, correlation_matrix: Array | None = None, correlation_cholesky: Array | None = None, *, validate_args: bool | None = None)[source]

Bases: Distribution

A distribution that links the batch_shape[:-1] of marginal distribution marginal_dist with a multivariate Gaussian copula modelling the correlation between the axes.

Parameters:

marginal_dist (Distribution) – Distribution whose last batch axis is to be coupled.
correlation_matrix (array_like) – Correlation matrix of coupling multivariate normal distribution.
correlation_cholesky (array_like) – Correlation Cholesky factor of coupling multivariate normal distribution.

arg_constraints: dict[str, Any] = {'correlation_cholesky': CorrCholesky(), 'correlation_matrix': CorrMatrix()}

reparametrized_params: list[str] = ['correlation_matrix', 'correlation_cholesky']

pytree_data_fields: tuple[str, ...] = ('marginal_dist', 'base_dist')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

correlation_matrix() → Array[source]

correlation_cholesky() → Array[source]

GaussianCopulaBeta

Bases: GaussianCopula

arg_constraints: dict[str, Any] = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0), 'correlation_cholesky': CorrCholesky(), 'correlation_matrix': CorrMatrix()}

support = IndependentConstraint(UnitInterval(lower_bound=0.0, upper_bound=1.0), 1)

pytree_data_fields: tuple[str, ...] = ('concentration1', 'concentration0')

GaussianRandomWalk

Bases: Distribution

arg_constraints: dict[str, Any] = {'scale': Positive(lower_bound=0.0)}

support = RealVector(Real(), 1)

reparametrized_params: list[str] = ['scale']

pytree_aux_fields: tuple[str, ...] = ('num_steps',)

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

GaussianStateSpace

class GaussianStateSpace(num_steps: int, transition_matrix: Array, covariance_matrix: Array | None = None, precision_matrix: Array | None = None, scale_tril: Array | None = None, initial_value: Array | None = None, *, validate_args: bool | None = None)[source]

Gaussian state space model.

\[\begin{split}\mathbf{z}_{t} &= \mathbf{A} \mathbf{z}_{t - 1} + \boldsymbol{\epsilon}_t\\ &= \mathbf{A}^t \mathbf{z}_0 + \sum_{k=1}^{t} \mathbf{A}^{t-k} \boldsymbol{\epsilon}_k,\end{split}\]

where \(\mathbf{z}_t\) is the state vector at step \(t\), \(\mathbf{A}\) is the transition matrix, \(\mathbf{z}_0\) is the initial value, and \(\boldsymbol\epsilon\) is the innovation noise.

Parameters:

num_steps – Number of steps.
transition_matrix – State space transition matrix \(\mathbf{A}\).
covariance_matrix – Covariance of the innovation noise \(\boldsymbol\epsilon\).
precision_matrix – Precision matrix of the innovation noise \(\boldsymbol\epsilon\).
scale_tril – Scale matrix of the innovation noise \(\boldsymbol\epsilon\).
initial_value – Initial state vector \(\mathbf{z}_0\). If None, defaults to zero.

arg_constraints: dict[str, Any] = {'covariance_matrix': PositiveDefinite(), 'initial_value': RealVector(Real(), 1), 'precision_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky(), 'transition_matrix': RealMatrix(Real(), 2)}

support = RealMatrix(Real(), 2)

pytree_data_fields: tuple[str, ...] = ('transition_matrix', '_initial_value', 'scale_tril')

pytree_aux_fields: tuple[str, ...] = ('num_steps',)

property initial_value: Array

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

covariance_matrix()[source]

precision_matrix()[source]

Gompertz

Bases: Distribution

Gompertz Distribution.

The Gompertz distribution is a distribution with support on the positive real line that is closely related to the Gumbel distribution. This implementation follows the notation used in the Wikipedia entry for the Gompertz distribution. See https://en.wikipedia.org/wiki/Gompertz_distribution.

However, we call the parameter “eta” a concentration parameter and the parameter “b” a rate parameter (as opposed to scale parameter as in wikipedia description.)

The CDF, in terms of concentration (con) and rate, is

\[F(x) = 1 - \exp \left\{ - \text{con} * \left [ \exp\{x * rate \} - 1 \right ] \right\}\]

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}

support = Positive(lower_bound=0.0)

reparametrized_params: list[str] = ['concentration', 'rate']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

Gumbel

Bases: Distribution

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Real()

reparametrized_params: list[str] = ['loc', 'scale']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

HalfCauchy

Bases: Distribution

reparametrized_params: list[str] = ['scale']

support = Positive(lower_bound=0.0)

arg_constraints: dict[str, Any] = {'scale': Positive(lower_bound=0.0)}

pytree_data_fields: tuple[str, ...] = ('_cauchy', 'scale')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

HalfNormal

Bases: Distribution

reparametrized_params: list[str] = ['scale']

support = Positive(lower_bound=0.0)

arg_constraints: dict[str, Any] = {'scale': Positive(lower_bound=0.0)}

pytree_data_fields: tuple[str, ...] = ('_normal', 'scale')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

InverseGamma

Note

We keep the same notation rate as in Pyro but it plays the role of scale parameter of InverseGamma in literatures (e.g. wikipedia: https://en.wikipedia.org/wiki/Inverse-gamma_distribution)

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['concentration', 'rate']

support = Positive(lower_bound=0.0)

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

InverseWishart

[1] https://en.wikipedia.org/wiki/Inverse-Wishart_distribution

Inverse Wishart distribution for covariance matrices.

The Inverse Wishart distribution is the conjugate prior for the covariance matrix of a multivariate normal distribution. If \(\mathbf{X} \sim W^{-1}(\mathbf{\Psi}, \nu)\), then \(\mathbf{X}^{-1} \sim W(\mathbf{\Psi}^{-1}, \nu)\) (Wishart distribution).

\[p(\mathbf{X} \mid \mathbf{\Psi}, \nu) = \frac{|\mathbf{\Psi}|^{\nu/2}}{2^{\nu p/2} \Gamma_p(\nu/2)} |\mathbf{X}|^{-(\nu + p + 1)/2} \exp\left( -\frac{1}{2} \mathrm{tr}(\mathbf{\Psi} \mathbf{X}^{-1}) \right)\]

where \(p\) is the dimension of the matrix, \(\nu > p - 1\) is the degrees of freedom, and \(\mathbf{\Psi}\) is the positive definite scale matrix.

Parameters:

concentration – Degrees of freedom parameter (often denoted \(\nu\)). Must be greater than p - 1 where p is the dimension of the scale matrix.
scale_matrix – Positive definite scale matrix \(\mathbf{\Psi}\), analogous to the inverse rate of a Gamma distribution.
rate_matrix – Inverse of the scale matrix, analogous to the rate of a Gamma distribution.
scale_tril – Cholesky decomposition of the scale matrix.

Properties

Mean: \(\frac{\mathbf{\Psi}}{\nu - p - 1}\) for \(\nu > p + 1\)
Mode: \(\frac{\mathbf{\Psi}}{\nu + p + 1}\)

References

arg_constraints: dict[str, Any] = {'concentration': Dependent(), 'rate_matrix': PositiveDefinite(), 'scale_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}

support = PositiveDefinite()

reparametrized_params: list[str] = ['scale_matrix', 'rate_matrix', 'scale_tril']

concentration()[source]

scale_matrix()[source]

rate_matrix()[source]

scale_tril()[source]

mean() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Mean of the distribution.

mode() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

variance() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Variance of the distribution.

static infer_shapes(concentration=(), scale_matrix=None, rate_matrix=None, scale_tril=None)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

[1] https://en.wikipedia.org/wiki/Inverse-Wishart_distribution

InverseWishartCholesky

Bases: Distribution

Cholesky factor of an Inverse Wishart distribution for covariance matrices.

This distribution samples the Cholesky factor \(\mathbf{L}\) such that \(\mathbf{X} = \mathbf{L} \mathbf{L}^T \sim W^{-1}(\mathbf{\Psi}, \nu)\).

Parameters:

concentration – Degrees of freedom parameter (often denoted \(\nu\)). Must be greater than p - 1 where p is the dimension of the scale matrix.
scale_matrix – Positive definite scale matrix \(\mathbf{\Psi}\), analogous to the inverse rate of a Gamma distribution.
rate_matrix – Inverse of the scale matrix, analogous to the rate of a Gamma distribution.
scale_tril – Cholesky decomposition of the scale matrix.

References

arg_constraints: dict[str, Any] = {'concentration': Dependent(), 'rate_matrix': PositiveDefinite(), 'scale_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}

support = LowerCholesky()

reparametrized_params: list[str] = ['scale_matrix', 'rate_matrix', 'scale_tril']

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

scale_matrix()[source]

rate_matrix()[source]

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

mean() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Mean of the distribution.

variance() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Variance of the distribution.

static infer_shapes(concentration: tuple[int, ...] = (), scale_matrix: tuple[int, ...] | None = None, rate_matrix: tuple[int, ...] | None = None, scale_tril: tuple[int, ...] | None = None)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

Kumaraswamy

Bases: Distribution

arg_constraints: dict[str, Any] = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['concentration1', 'concentration0']

support = UnitInterval(lower_bound=0.0, upper_bound=1.0)

KL_KUMARASWAMY_BETA_TAYLOR_ORDER = 10

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

Laplace

Bases: Distribution

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Real()

reparametrized_params: list[str] = ['loc', 'scale']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Levy

Bases: Distribution

Lévy distribution is a special case of Lévy alpha-stable distribution. Its probability density function is given by,

\[f(x\mid \mu, c) = \sqrt{\frac{c}{2\pi(x-\mu)^{3}}} \exp\left(-\frac{c}{2(x-\mu)}\right), \qquad x > \mu\]

where \(\mu\) is the location parameter and \(c\) is the scale parameter.

Parameters:

loc – Location parameter.
scale – Scale parameter.

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Compute the log probability density function of the Lévy distribution.

\[\log f(x\mid \mu, c) = \frac{1}{2}\log\left(\frac{c}{2\pi}\right) - \frac{c}{2(x-\mu)} - \frac{3}{2}\log(x-\mu), \qquad x > \mu\]

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: numpy.ndarray

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

The inverse cumulative distribution function of Lévy distribution is given by,

\[F^{-1}(q\mid \mu, c) = \mu + c\left(\Phi^{-1}(1-q/2)\right)^{-2}\]

where \(\Phi^{-1}\) is the inverse of the standard normal cumulative distribution function.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

The cumulative distribution function of Lévy distribution is given by,

\[F(x\mid \mu, c) = 2 - 2\Phi\left(\sqrt{\frac{c}{x-\mu}}\right)\]

where \(\Phi\) is the standard normal cumulative distribution function.

Parameters:: value – samples from Lévy distribution.
Returns:: output of the cumulative distribution function evaluated at value.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: If \(X \sim \text{Levy}(\mu, c)\), then the entropy of \(X\) is given by,

\[H(X) = \frac{1}{2}+\frac{3}{2}\gamma+\frac{1}{2}\ln{\left(16\pi c^2\right)}\]

LKJ

LKJ distribution for correlation matrices. The distribution is controlled by concentration parameter \(\eta\) to make the probability of the correlation matrix \(M\) proportional to \(\det(M)^{\eta - 1}\). Because of that, when concentration == 1, we have a uniform distribution over correlation matrices.

When concentration > 1, the distribution favors samples with large large determinent. This is useful when we know a priori that the underlying variables are not correlated.

When concentration < 1, the distribution favors samples with small determinent. This is useful when we know a priori that some underlying variables are correlated.

Sample code for using LKJ in the context of multivariate normal sample:

def model(y):  # y has dimension N x d
    d = y.shape[1]
    N = y.shape[0]
    # Vector of variances for each of the d variables
    theta = numpyro.sample("theta", dist.HalfCauchy(jnp.ones(d)))

    concentration = jnp.ones(1)  # Implies a uniform distribution over correlation matrices
    corr_mat = numpyro.sample("corr_mat", dist.LKJ(d, concentration))
    sigma = jnp.sqrt(theta)
    # we can also use a faster formula `cov_mat = jnp.outer(sigma, sigma) * corr_mat`
    cov_mat = jnp.matmul(jnp.matmul(jnp.diag(sigma), corr_mat), jnp.diag(sigma))

    # Vector of expectations
    mu = jnp.zeros(d)

    with numpyro.plate("observations", N):
        obs = numpyro.sample("obs", dist.MultivariateNormal(mu, covariance_matrix=cov_mat), obs=y)
    return obs

Parameters:

dimension (int) – dimension of the matrices
concentration (ndarray) – concentration/shape parameter of the distribution (often referred to as eta)
sample_method (str) – Either “cvine” or “onion”. Both methods are proposed in [1] and offer the same distribution over correlation matrices. But they are different in how to generate samples. Defaults to “onion”.

References

[1] Generating random correlation matrices based on vines and extended onion method, Daniel Lewandowski, Dorota Kurowicka, Harry Joe

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['concentration']

support = CorrMatrix()

pytree_aux_fields: tuple[str, ...] = ('dimension', 'sample_method')

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

LKJCholesky

Bases: Distribution

LKJ distribution for lower Cholesky factors of correlation matrices. The distribution is controlled by concentration parameter \(\eta\) to make the probability of the correlation matrix \(M\) generated from a Cholesky factor proportional to \(\det(M)^{\eta - 1}\). Because of that, when concentration == 1, we have a uniform distribution over Cholesky factors of correlation matrices.

When concentration > 1, the distribution favors samples with large diagonal entries (hence large determinent). This is useful when we know a priori that the underlying variables are not correlated.

When concentration < 1, the distribution favors samples with small diagonal entries (hence small determinent). This is useful when we know a priori that some underlying variables are correlated.

Sample code for using LKJCholesky in the context of multivariate normal sample:

def model(y):  # y has dimension N x d
    d = y.shape[1]
    N = y.shape[0]
    # Vector of variances for each of the d variables
    theta = numpyro.sample("theta", dist.HalfCauchy(jnp.ones(d)))
    # Lower cholesky factor of a correlation matrix
    concentration = jnp.ones(1)  # Implies a uniform distribution over correlation matrices
    L_omega = numpyro.sample("L_omega", dist.LKJCholesky(d, concentration))
    # Lower cholesky factor of the covariance matrix
    sigma = jnp.sqrt(theta)
    # we can also use a faster formula `L_Omega = sigma[..., None] * L_omega`
    L_Omega = jnp.matmul(jnp.diag(sigma), L_omega)

    # Vector of expectations
    mu = jnp.zeros(d)

    with numpyro.plate("observations", N):
        obs = numpyro.sample("obs", dist.MultivariateNormal(mu, scale_tril=L_Omega), obs=y)
    return obs

Parameters:

dimension (int) – dimension of the matrices
concentration (ndarray) – concentration/shape parameter of the distribution (often referred to as eta)
sample_method (str) – Either “cvine” or “onion”. Both methods are proposed in [1] and offer the same distribution over correlation matrices. But they are different in how to generate samples. Defaults to “onion”.

References

[1] Generating random correlation matrices based on vines and extended onion method, Daniel Lewandowski, Dorota Kurowicka, Harry Joe

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['concentration']

support = CorrCholesky()

pytree_data_fields: tuple[str, ...] = ('_beta', 'concentration')

pytree_aux_fields: tuple[str, ...] = ('dimension', 'sample_method')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

LogNormal

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Positive(lower_bound=0.0)

reparametrized_params: list[str] = ['loc', 'scale']

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

LogUniform

arg_constraints: dict[str, Any] = {'high': Positive(lower_bound=0.0), 'low': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['low', 'high']

pytree_data_fields: tuple[str, ...] = ('low', 'high', '_support')

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Logistic

Bases: Distribution

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Real()

reparametrized_params: list[str] = ['loc', 'scale']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

LowRankMultivariateNormal

class LowRankMultivariateNormal(loc: Array, cov_factor: Array, cov_diag: Array, *, validate_args: bool | None = None)[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {'cov_diag': IndependentConstraint(Positive(lower_bound=0.0), 1), 'cov_factor': IndependentConstraint(Real(), 2), 'loc': RealVector(Real(), 1)}

support = RealVector(Real(), 1)

reparametrized_params: list[str] = ['loc', 'cov_factor', 'cov_diag']

pytree_data_fields: tuple[str, ...] = ('loc', 'cov_factor', 'cov_diag', '_capacitance_tril')

property mean: Array: Mean of the distribution.

variance() → Array[source]: Variance of the distribution.

scale_tril() → Array[source]

covariance_matrix() → Array[source]

precision_matrix() → Array[source]

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

static infer_shapes(loc, cov_factor, cov_diag)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

[1] https://en.wikipedia.org/wiki/Matrix_normal_distribution

MatrixNormal

class MatrixNormal(loc: Array, scale_tril_row: Array, scale_tril_column: Array, *, validate_args: bool | None = None)[source]

Bases: Distribution

Matrix variate normal distribution as described in [1] but with a lower_triangular parametrization, i.e. \(U=scale_tril_row @ scale_tril_row^{T}\) and \(V=scale_tril_column @ scale_tril_column^{T}\). The distribution is related to the multivariate normal distribution in the following way. If \(X ~ MN(loc,U,V)\) then \(vec(X) ~ MVN(vec(loc), kron(V,U) )\).

Parameters:

loc (array_like) – Location of the distribution.
scale_tril_row (array_like) – Lower cholesky of rows covariance matrix.
scale_tril_column (array_like) – Lower cholesky of columns covariance matrix.

References

arg_constraints: dict[str, Any] = {'loc': RealVector(Real(), 1), 'scale_tril_column': LowerCholesky(), 'scale_tril_row': LowerCholesky()}

support = RealMatrix(Real(), 2)

reparametrized_params: list[str] = ['loc', 'scale_tril_row', 'scale_tril_column']

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

log_prob(values)[source]

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

MultivariateNormal

Bases: Distribution

arg_constraints: dict[str, Any] = {'covariance_matrix': PositiveDefinite(), 'loc': RealVector(Real(), 1), 'precision_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}

support = RealVector(Real(), 1)

reparametrized_params: list[str] = ['loc', 'covariance_matrix', 'precision_matrix', 'scale_tril']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

covariance_matrix()[source]

precision_matrix()[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

static infer_shapes(loc=(), covariance_matrix=None, precision_matrix=None, scale_tril=None)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

MultivariateStudentT

Bases: Distribution

arg_constraints: dict[str, Any] = {'df': Positive(lower_bound=0.0), 'loc': RealVector(Real(), 1), 'scale_tril': LowerCholesky()}

support = RealVector(Real(), 1)

reparametrized_params: list[str] = ['df', 'loc', 'scale_tril']

pytree_data_fields: tuple[str, ...] = ('df', 'loc', 'scale_tril', '_chi2')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

covariance_matrix()[source]

precision_matrix() → Array[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

static infer_shapes(df, loc, scale_tril)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

Normal

Bases: Distribution

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Real()

reparametrized_params: list[str] = ['loc', 'scale']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

log_cdf(value: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Pareto

arg_constraints: dict[str, Any] = {'alpha': Positive(lower_bound=0.0), 'scale': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['scale', 'alpha']

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

RelaxedBernoulli

RelaxedBernoulli(temperature, probs=None, logits=None, *, validate_args: bool | None = None)[source]

RelaxedBernoulliLogits

arg_constraints: dict[str, Any] = {'logits': Real(), 'temperature': Positive(lower_bound=0.0)}

support = UnitInterval(lower_bound=0.0, upper_bound=1.0)

SoftLaplace

Bases: Distribution

Smooth distribution with Laplace-like tail behavior.

This distribution corresponds to the log-convex density:

z = (value - loc) / scale
log_prob = log(2 / pi) - log(scale) - logaddexp(z, -z)

Like the Laplace density, this density has the heaviest possible tails (asymptotically) while still being log-convex. Unlike the Laplace distribution, this distribution is infinitely differentiable everywhere, and is thus suitable for HMC and Laplace approximation.

Parameters:

loc – Location parameter.
scale – Scale parameter.

arg_constraints: dict[str, Any] = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Real()

reparametrized_params: list[str] = ['loc', 'scale']

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

StudentT

Bases: Distribution

arg_constraints: dict[str, Any] = {'df': Positive(lower_bound=0.0), 'loc': Real(), 'scale': Positive(lower_bound=0.0)}

support = Real()

reparametrized_params: list[str] = ['df', 'loc', 'scale']

pytree_data_fields: tuple[str, ...] = ('df', 'loc', 'scale', '_chi2')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Uniform

Bases: Distribution

arg_constraints: dict[str, Any] = {'high': Dependent(), 'low': Dependent()}

reparametrized_params: list[str] = ['low', 'high']

pytree_data_fields: tuple[str, ...] = ('low', 'high', '_support')

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

static infer_shapes(low: tuple[int, ...] = (), high: tuple[int, ...] = ()) → tuple[tuple[int, ...], tuple[int, ...]][source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Weibull

Bases: Distribution

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'scale': Positive(lower_bound=0.0)}

support = Positive(lower_bound=0.0)

reparametrized_params: list[str] = ['scale', 'concentration']

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Wishart

Wishart distribution for covariance matrices.

Parameters:

concentration – Positive concentration parameter analogous to the concentration of a Gamma distribution. The concentration must be larger than the dimensionality of the scale matrix.
scale_matrix – Scale matrix analogous to the inverse rate of a Gamma distribution.
rate_matrix – Rate matrix anaologous to the rate of a Gamma distribution.
scale_tril – Cholesky decomposition of the scale_matrix.

arg_constraints: dict[str, Any] = {'concentration': Dependent(), 'rate_matrix': PositiveDefinite(), 'scale_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}

support = PositiveDefinite()

reparametrized_params: list[str] = ['scale_matrix', 'rate_matrix', 'scale_tril']

concentration()[source]

scale_matrix()[source]

rate_matrix()[source]

scale_tril()[source]

mean() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Mean of the distribution.

variance() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Variance of the distribution.

static infer_shapes(concentration=(), scale_matrix=None, rate_matrix=None, scale_tril=None)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

WishartCholesky

Bases: Distribution

Cholesky factor of a Wishart distribution for covariance matrices.

Parameters:

concentration – Positive concentration parameter analogous to the concentration of a Gamma distribution. The concentration must be larger than the dimensionality of the scale matrix.
scale_matrix – Scale matrix analogous to the inverse rate of a Gamma distribution.
rate_matrix – Rate matrix anaologous to the rate of a Gamma distribution.
scale_tril – Cholesky decomposition of the scale_matrix.

arg_constraints: dict[str, Any] = {'concentration': Dependent(), 'rate_matrix': PositiveDefinite(), 'scale_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}

support = LowerCholesky()

reparametrized_params: list[str] = ['scale_matrix', 'rate_matrix', 'scale_tril']

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

scale_matrix()[source]

rate_matrix()[source]

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

mean() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Mean of the distribution.

variance() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Variance of the distribution.

static infer_shapes(concentration: tuple[int, ...] = (), scale_matrix: tuple[int, ...] | None = None, rate_matrix: tuple[int, ...] | None = None, scale_tril: tuple[int, ...] | None = None)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

ZeroSumNormal

References [1] https://github.com/pymc-devs/pymc/blob/6252d2e58dc211c913ee2e652a4058d271d48bbd/pymc/distributions/multivariate.py#L2637 [2] https://www.pymc.io/projects/docs/en/stable/api/distributions/generated/pymc.ZeroSumNormal.html [3] https://learnbayesstats.com/episode/74-optimizing-nuts-developing-zerosumnormal-distribution-adrian-seyboldt/

Zero Sum Normal distribution adapted from PyMC [1] as described in [2,3]. This is a Normal distribution where one or more axes are constrained to sum to zero (the last axis by default).

\[\begin{split}\begin{align*} ZSN(\sigma) = N(0, \sigma^2 (I - \tfrac{1}{n}J)) \\ \text{where} \ ~ J_{ij} = 1 \ ~ \text{and} \\ n = \text{number of zero-sum axes} \end{align*}\end{split}\]

Parameters:

scale (array_like) – Standard deviation of the underlying normal distribution before the zerosum constraint is enforced.
event_shape (tuple) – The event shape of the distribution, the axes of which get constrained to sum to zero.

Example:

>>> from numpy.testing import assert_allclose
>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer import MCMC, NUTS

>>> N = 1000
>>> n_categories = 20
>>> rng_key = random.key(0)
>>> key1, key2, key3 = random.split(rng_key, 3)
>>> category_ind = random.choice(key1, jnp.arange(n_categories), shape=(N,))
>>> beta = random.normal(key2, shape=(n_categories,))
>>> beta -= beta.mean(-1)
>>> y = 5 + beta[category_ind] + random.normal(key3, shape=(N,))

>>> def model(category_ind, y): # category_ind is an indexed categorical variable with 20 categories
...     N = len(category_ind)
...     alpha = numpyro.sample("alpha", dist.Normal(0, 2.5))
...     beta = numpyro.sample("beta", dist.ZeroSumNormal(1, event_shape=(n_categories,)))
...     sigma =  numpyro.sample("sigma", dist.Exponential(1))
...     with numpyro.plate("observations", N):
...         mu = alpha + beta[category_ind]
...         obs = numpyro.sample("obs", dist.Normal(mu, sigma), obs=y)
...     return obs

>>> nuts_kernel = NUTS(model=model, target_accept_prob=0.9)
>>> mcmc = MCMC(
...     sampler=nuts_kernel,
...     num_samples=1_000, num_warmup=1_000, num_chains=4
... )
>>> mcmc.run(random.key(0), category_ind=category_ind, y=y)
>>> posterior_samples = mcmc.get_samples()
>>> # Confirm everything along last axis sums to zero
>>> assert_allclose(posterior_samples['beta'].sum(-1), 0, atol=1e-3)

arg_constraints: dict[str, Any] = {'scale': Positive(lower_bound=0.0)}

reparametrized_params: list[str] = ['scale']

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

Discrete Distributions

Bernoulli

Bernoulli(probs: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, logits: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, *, validate_args: bool | None = None) → BernoulliProbs | BernoulliLogits[source]

BernoulliLogits

Bases: Distribution

arg_constraints: dict[str, Any] = {'logits': Real()}

support = Boolean()

has_enumerate_support: bool = True

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

probs() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

BernoulliProbs

Bases: Distribution

arg_constraints: dict[str, Any] = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0)}

support = Boolean()

has_enumerate_support: bool = True

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

logits() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

BetaBinomial

Bases: Distribution

Compound distribution comprising of a beta-binomial pair. The probability of success (probs for the Binomial distribution) is unknown and randomly drawn from a Beta distribution prior to a certain number of Bernoulli trials given by total_count.

Parameters:

concentration1 (numpy.ndarray) – 1st concentration parameter (alpha) for the Beta distribution.
concentration0 (numpy.ndarray) – 2nd concentration parameter (beta) for the Beta distribution.
total_count (numpy.ndarray) – number of Bernoulli trials.

arg_constraints: dict[str, Any] = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0), 'total_count': IntegerNonnegative(lower_bound=0)}

has_enumerate_support: bool = True

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Returns an array with shape len(support) x batch_shape containing all values in the support.

pytree_data_fields: tuple[str, ...] = ('concentration1', 'concentration0', 'total_count', '_beta')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

[1] https://en.wikipedia.org/wiki/Beta_negative_binomial_distribution [2] https://mc-stan.org/docs/functions-reference/unbounded_discrete_distributions.html#beta-neg-binomial

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

BetaNegativeBinomial

Bases: Distribution

Compound distribution comprising of a beta-negative-binomial pair. The probs parameter for the NegativeBinomialProbs distribution is unknown and randomly drawn from a Beta distribution prior to the negative binomial counting process.

The Beta Negative Binomial is a heavy-tailed discrete distribution useful for modeling overdispersed count data. It arises as the marginal distribution when integrating out the success probability in a negative binomial model with a beta prior.

Parameters:

concentration1 (numpy.ndarray) – 1st concentration parameter (alpha) for the Beta distribution.
concentration0 (numpy.ndarray) – 2nd concentration parameter (beta) for the Beta distribution.
n (numpy.ndarray) – positive number of successes parameter for the negative binomial distribution.

References

arg_constraints: dict[str, Any] = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0), 'n': Positive(lower_bound=0.0)}

support = IntegerNonnegative(lower_bound=0)

pytree_data_fields: tuple[str, ...] = ('concentration1', 'concentration0', 'n', '_beta')

If \(X \sim \mathrm{BetaNegativeBinomial}(\alpha, \beta, n)\), then the sampling procedure is:

\[\begin{split}\begin{align*} p &\sim \mathrm{Beta}(\alpha, \beta) \\ X \mid p &\sim \mathrm{NegativeBinomial}(n, p) \end{align*}\end{split}\]

It uses Beta to generate samples from the Beta distribution and NegativeBinomialProbs to generate samples from the Negative Binomial distribution.

If \(X \sim \mathrm{BetaNegativeBinomial}(\alpha, \beta, n)\), then the log probability mass function is:

\[P(X = k) = \binom{n + k - 1}{k} \frac{B(\alpha + k, \beta + n)}{B(\alpha, \beta)}\]

To ensure differentiability, the binomial coefficient is computed using gamma functions.

If \(X \sim \mathrm{BetaNegativeBinomial}(\alpha, \beta, n)\) and \(\beta > 1\), then the mean is:

\[\mathbb{E}[X] = \frac{n\alpha}{\beta - 1},\]

otherwise, the its undefined.

If \(X \sim \mathrm{BetaNegativeBinomial}(\alpha, \beta, n)\) and \(\beta > 2\), then the variance is:

\[\mathrm{Var}[X] = \frac{n\alpha (n + \beta - 1)(\alpha + \beta - 1)}{(\beta - 1)^2 \cdot (\beta - 2)},\]

otherwise, the its undefined.

Binomial

Binomial(total_count: int = 1, probs: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, logits: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, *, validate_args: bool | None = None) → BinomialProbs | BinomialLogits[source]

BinomialLogits

Bases: Distribution

arg_constraints: dict[str, Any] = {'logits': Real(), 'total_count': IntegerNonnegative(lower_bound=0)}

has_enumerate_support: bool = True

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Returns an array with shape len(support) x batch_shape containing all values in the support.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

probs() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

BinomialProbs

Bases: Distribution

arg_constraints: dict[str, Any] = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0), 'total_count': IntegerNonnegative(lower_bound=0)}

has_enumerate_support: bool = True

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

logits() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

Categorical

Categorical(probs=None, logits=None, *, validate_args: bool | None = None)[source]

CategoricalLogits

class CategoricalLogits(logits: Array, *, validate_args: bool | None = None)[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {'logits': RealVector(Real(), 1)}

has_enumerate_support: bool = True

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

probs() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

CategoricalProbs

class CategoricalProbs(probs: Array, *, validate_args: bool | None = None)[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {'probs': Simplex()}

has_enumerate_support: bool = True

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

logits() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

DirichletMultinomial

Bases: Distribution

Compound distribution comprising of a dirichlet-multinomial pair. The probability of classes (probs for the Multinomial distribution) is unknown and randomly drawn from a Dirichlet distribution prior to a certain number of Categorical trials given by total_count.

Parameters:

concentration (numpy.ndarray) – concentration parameter (alpha) for the Dirichlet distribution.
total_count (numpy.ndarray) – number of Categorical trials.
total_count_max (int) – the maximum number of trials, i.e. max(total_count)

arg_constraints: dict[str, Any] = {'concentration': IndependentConstraint(Positive(lower_bound=0.0), 1), 'total_count': IntegerNonnegative(lower_bound=0)}

pytree_data_fields: tuple[str, ...] = ('concentration', '_dirichlet')

pytree_aux_fields: tuple[str, ...] = ('total_count', 'total_count_max')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

DiscreteUniform

Bases: Distribution

arg_constraints: dict[str, Any] = {'high': Dependent(), 'low': Dependent()}

has_enumerate_support: bool = True

pytree_data_fields: tuple[str, ...] = ('low', 'high', '_support')

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

enumerate_support(expand: bool = True) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns an array with shape len(support) x batch_shape containing all values in the support.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

GammaPoisson

Bases: Distribution

Compound distribution comprising of a gamma-poisson pair, also referred to as a gamma-poisson mixture. The rate parameter for the Poisson distribution is unknown and randomly drawn from a Gamma distribution.

Parameters:

concentration (numpy.ndarray) – shape parameter (alpha) of the Gamma distribution.
rate (numpy.ndarray) – rate parameter (rate) for the Gamma distribution.

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}

support = IntegerNonnegative(lower_bound=0)

pytree_data_fields: tuple[str, ...] = ('concentration', 'rate', '_gamma')

If \(X \sim \mathrm{GammaPoisson}(\alpha, \lambda)\), then the sampling procedure is:

\[\begin{split}\begin{align*} \theta &\sim \mathrm{Gamma}(\alpha, \lambda) \\ X \mid \theta &\sim \mathrm{Poisson}(\theta) \end{align*}\end{split}\]

It uses Gamma to generate samples from the Gamma distribution and Poisson to generate samples from the Poisson distribution.

log_prob(value: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: If \(X \sim \mathrm{GammaPoisson}(\alpha, \lambda)\), then the log probability mass function is:

\[p_{X}(k) = \frac{\lambda^\alpha}{(\alpha + k)(1+\lambda)^{\alpha + k}\mathrm{B}(\alpha, k + 1)}\]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: If \(X \sim \mathrm{GammaPoisson}(\alpha, \lambda)\), then the mean is:

\[\mathbb{E}[X] = \frac{\alpha}{\lambda}\]

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: If \(X \sim \mathrm{GammaPoisson}(\alpha, \lambda)\), then the variance is:

\[\mathrm{Var}[X] = \frac{\alpha}{\lambda^2}(1 + \lambda)\]

If \(X \sim \mathrm{GammaPoisson}(\alpha, \lambda)\), then the cumulative distribution function is:

\[F_{X}(x) = \frac{1}{\mathrm{B}(\alpha, x + 1)} \int_{0}^{\frac{\lambda}{1 + \lambda}} t^{\alpha - 1} (1 - t)^{x} dt\]

which is the regularized incomplete beta function. This implementation uses betainc().

Geometric

Geometric(probs: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, logits: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, *, validate_args: bool | None = None) → GeometricProbs | GeometricLogits[source]

GeometricLogits

Bases: Distribution

arg_constraints: dict[str, Any] = {'logits': Real()}

support = IntegerNonnegative(lower_bound=0)

probs() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

GeometricProbs

Bases: Distribution

arg_constraints: dict[str, Any] = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0)}

support = IntegerNonnegative(lower_bound=0)

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

logits() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

Multinomial

Multinomial(total_count=1, probs: Array = None, logits: Array = None, *, total_count_max: int | None = None, validate_args: bool | None = None) → MultinomialProbs | MultinomialLogits[source]

Multinomial distribution.

Parameters:

total_count – number of trials. If this is a JAX array, it is required to specify total_count_max.
probs – event probabilities
logits – event log probabilities
total_count_max (int) – the maximum number of trials, i.e. max(total_count)

MultinomialLogits

class MultinomialLogits(logits: Array, total_count: int = 1, *, total_count_max: int | None = None, validate_args: bool | None = None)[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {'logits': RealVector(Real(), 1), 'total_count': IntegerNonnegative(lower_bound=0)}

pytree_data_fields: tuple[str, ...] = ('logits',)

pytree_aux_fields: tuple[str, ...] = ('total_count', 'total_count_max')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

probs() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

static infer_shapes(logits: Array, total_count: int) → tuple[tuple[int, ...], tuple[int, ...]][source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

MultinomialProbs

class MultinomialProbs(probs: Array, total_count: int = 1, *, total_count_max: int | None = None, validate_args: bool | None = None)[source]

Bases: Distribution

arg_constraints: dict[str, Any] = {'probs': Simplex(), 'total_count': IntegerNonnegative(lower_bound=0)}

pytree_data_fields: tuple[str, ...] = ('probs',)

pytree_aux_fields: tuple[str, ...] = ('total_count', 'total_count_max')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

logits() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

static infer_shapes(probs: Array, total_count: int) → tuple[tuple[int, ...], tuple[int, ...]][source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

OrderedLogistic

Bases: CategoricalProbs

A categorical distribution with ordered outcomes.

References:

Stan Functions Reference, v2.20 section 12.6, Stan Development Team

Parameters:

predictor (numpy.ndarray) – prediction in real domain; typically this is output of a linear model.
cutpoints (numpy.ndarray) – positions in real domain to separate categories.

arg_constraints: dict[str, Any] = {'cutpoints': OrderedVector(), 'predictor': Real()}

static infer_shapes(predictor, cutpoints)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

entropy() → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: Returns the entropy of the distribution.

NegativeBinomial

Factory function for Negative Binomial distribution.

Parameters:

total_count (int) – Number of successful trials.
probs (Optional[ArrayLike]) – Probability of success for each trial, by default None
logits (Optional[ArrayLike]) – Log-odds of success for each trial, by default None
validate_args (Optional[bool]) – Whether to validate the parameters, by default None

Returns:

An instance of NegativeBinomialProbs or NegativeBinomialLogits depending on the provided parameters.

Return type:

GammaPoisson

Raises:

ValueError – If neither probs nor logits is specified.

NegativeBinomialLogits

Bases: GammaPoisson

Negative Binomial distribution parameterized by total_count (\(r\)) and logits (\(\displaystyle\mathrm{logits}(p)=\log \frac{p}{1-p}\)). It is implemented as a \(\mathrm{GammaPoisson}(n, \exp(-\mathrm{logits}(p)))\) distribution.

Parameters:

total_count – Number of successful trials.
logits – Log-odds of success for each trial (\(\ln \frac{p}{1-p}\)).

arg_constraints: dict[str, Any] = {'logits': Real(), 'total_count': Positive(lower_bound=0.0)}

support = IntegerNonnegative(lower_bound=0)

log_prob(value: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]: If \(X \sim \mathrm{NegativeBinomial}(r, \mathrm{logits}(p))\), then the log probability mass function is:

\[\ln P(X = k) = -r \ln(1+\exp(\mathrm{logits}(p))) - k \ln(1+\exp(-\mathrm{logits}(p))) - \ln\Gamma(1 + k) - \ln\Gamma(\alpha) + \ln\Gamma(k + \alpha)\]

NegativeBinomialProbs

Bases: GammaPoisson

Negative Binomial distribution parameterized by total_count (\(r\)) and probs (\(p\)). It is implemented as a \(\displaystyle\mathrm{GammaPoisson}(n, \frac{1}{p} - 1)\) distribution.

Parameters:

total_count – Number of successful trials (\(r\)).
probs – Probability of success for each trial (\(p\)).

arg_constraints: dict[str, Any] = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0), 'total_count': Positive(lower_bound=0.0)}

support = IntegerNonnegative(lower_bound=0)

NegativeBinomial2

Bases: GammaPoisson

If \(X \sim \mathrm{NegativeBinomial2}(\mu, \alpha)\), then \(X \sim \mathrm{GammaPoisson}(\alpha, \frac{\alpha}{\mu})\).

Parameters:

mean (numpy.ndarray) – mean parameter (\(\mu\)).
concentration (numpy.ndarray) – concentration parameter (\(\alpha\)).

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'mean': Positive(lower_bound=0.0)}

support = IntegerNonnegative(lower_bound=0)

pytree_data_fields: tuple[str, ...] = ('concentration',)

Poisson

Bases: Distribution

Creates a Poisson distribution parameterized by rate, the rate parameter.

Samples are nonnegative integers, with a pmf given by

\[\mathrm{rate}^k \frac{e^{-\mathrm{rate}}}{k!}\]

Parameters:

rate (numpy.ndarray) – The rate parameter
is_sparse (bool) – Whether to assume value is mostly zero when computing log_prob(), which can speed up computation when data is sparse.

arg_constraints: dict[str, Any] = {'rate': Positive(lower_bound=0.0)}

support = IntegerNonnegative(lower_bound=0)

pytree_aux_fields: tuple[str, ...] = ('is_sparse',)

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Variance of the distribution.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

ZeroInflatedDistribution

Generic Zero Inflated distribution.

Parameters:

base_dist (Distribution) – the base distribution.
gate (numpy.ndarray) – probability of extra zeros given via a Bernoulli distribution.
gate_logits (numpy.ndarray) – logits of extra zeros given via a Bernoulli distribution.

ZeroInflatedPoisson

Bases: ZeroInflatedProbs

A Zero Inflated Poisson distribution.

Parameters:

gate (numpy.ndarray) – probability of extra zeros.
rate (numpy.ndarray) – rate of Poisson distribution.

arg_constraints: dict[str, Any] = {'gate': UnitInterval(lower_bound=0.0, upper_bound=1.0), 'rate': Positive(lower_bound=0.0)}

support = IntegerNonnegative(lower_bound=0)

pytree_data_fields: tuple[str, ...] = ('rate',)

ZeroInflatedNegativeBinomial2

ZeroInflatedNegativeBinomial2(mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray, concentration: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray, *, gate: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, gate_logits: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, validate_args: bool | None = None)[source]

Mixture Distributions

Mixture

Mixture(mixing_distribution: CategoricalProbs | CategoricalLogits, component_distributions: list[Distribution] | Distribution, *, validate_args: bool | None = None)[source]

A marginalized finite mixture of component distributions

The returned distribution will be either a:

MixtureGeneral, when component_distributions is a list, or
MixtureSameFamily, when component_distributions is a single distribution.

and more details can be found in the documentation for each of these classes.

Parameters:

mixing_distribution – A Categorical specifying the weights for each mixture components. The size of this distribution specifies the number of components in the mixture, mixture_size.
component_distributions – Either a list of component distributions or a single vectorized distribution. When a list is provided, the number of elements must equal mixture_size. Otherwise, the last batch dimension of the distribution must equal mixture_size.

Returns:

The mixture distribution.

MixtureSameFamily

class MixtureSameFamily(mixing_distribution: CategoricalProbs | CategoricalLogits, component_distribution: Distribution, *, validate_args: bool | None = None)[source]

Bases: _MixtureBase

A finite mixture of component distributions from the same family

This mixture only supports a mixture of component distributions that are all of the same family. The different components are specified along the last batch dimension of the input component_distribution. If you need a mixture of distributions from different families, use the more general implementation in MixtureGeneral.

Parameters:

mixing_distribution – A Categorical specifying the weights for each mixture components. The size of this distribution specifies the number of components in the mixture, mixture_size.
component_distribution – A single vectorized Distribution, whose last batch dimension equals mixture_size as specified by mixing_distribution.

Example

>>> import jax
>>> import jax.numpy as jnp
>>> import numpyro.distributions as dist
>>> mixing_dist = dist.Categorical(probs=jnp.ones(3) / 3.)
>>> component_dist = dist.Normal(loc=jnp.zeros(3), scale=jnp.ones(3))
>>> mixture = dist.MixtureSameFamily(mixing_dist, component_dist)
>>> mixture.sample(jax.random.key(42)).shape
()

pytree_data_fields: tuple[str, ...] = ('_mixing_distribution', '_component_distribution')

pytree_aux_fields: tuple[str, ...] = ('_mixture_size',)

property component_distribution: Distribution

Return the vectorized distribution of components being mixed.

Returns:: Component distribution
Return type:: Distribution

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

property is_discrete: bool

property component_mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray

property component_variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray

component_cdf(samples: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

component_sample(key: prng_key, sample_shape: tuple[int, ...] = ()) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

component_log_probs(value: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

MixtureGeneral

class MixtureGeneral(mixing_distribution: CategoricalProbs | CategoricalLogits, component_distributions: list[Distribution], *, support: Constraint | None = None, validate_args: bool | None = None)[source]

Bases: _MixtureBase

A finite mixture of component distributions from different families

If all of the component distributions are from the same family, the more specific implementation in MixtureSameFamily will be somewhat more efficient.

Parameters:

mixing_distribution – A Categorical specifying the weights for each mixture components. The size of this distribution specifies the number of components in the mixture, mixture_size.
component_distributions – A list of mixture_size Distribution objects.
support – A Constraint object specifying the support of the mixture distribution. If not provided, the support will be inferred from the component distributions.

Example

>>> import jax
>>> import jax.numpy as jnp
>>> import numpyro.distributions as dist
>>> mixing_dist = dist.Categorical(probs=jnp.ones(3) / 3.)
>>> component_dists = [
...     dist.Normal(loc=0.0, scale=1.0),
...     dist.Normal(loc=-0.5, scale=0.3),
...     dist.Normal(loc=0.6, scale=1.2),
... ]
>>> mixture = dist.MixtureGeneral(mixing_dist, component_dists)
>>> mixture.sample(jax.random.key(42)).shape
()

>>> import jax
>>> import jax.numpy as jnp
>>> import numpyro.distributions as dist
>>> mixing_dist = dist.Categorical(probs=jnp.ones(2) / 2.)
>>> component_dists = [
...     dist.Normal(loc=0.0, scale=1.0),
...     dist.HalfNormal(scale=0.3),
... ]
>>> mixture = dist.MixtureGeneral(mixing_dist, component_dists, support=dist.constraints.real)
>>> mixture.sample(jax.random.key(42)).shape
()

pytree_data_fields: tuple[str, ...] = ('_mixing_distribution', '_component_distributions', '_support')

pytree_aux_fields: tuple[str, ...] = ('_mixture_size',)

property component_distributions: list[Distribution]

The list of component distributions in the mixture

Returns:: The list of component distributions
Return type:: list[Distribution]

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

property is_discrete: bool

property component_mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray

property component_variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray

component_cdf(samples: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray) → Array[source]

component_sample(key: prng_key, sample_shape: tuple[int, ...] = ()) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

component_log_probs(value: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray) → Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray[source]

Directional Distributions

ProjectedNormal

class ProjectedNormal(concentration: Array, *, validate_args: bool | None = None)[source]

Bases: Distribution

Projected isotropic normal distribution of arbitrary dimension.

This distribution over directional data is qualitatively similar to the von Mises and von Mises-Fisher distributions, but permits tractable variational inference via reparametrized gradients.

To use this distribution with autoguides and HMC, use handlers.reparam with a ProjectedNormalReparam reparametrizer in the model, e.g.:

@handlers.reparam(config={"direction": ProjectedNormalReparam()})
def model():
    direction = numpyro.sample("direction",
                               ProjectedNormal(zeros(3)))
    ...

Note

This implements log_prob() only for dimensions {2,3}.

[1] D. Hernandez-Stumpfhauser, F.J. Breidt, M.J. van der Woerd (2017): “The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference” https://projecteuclid.org/euclid.ba/1453211962

arg_constraints: dict[str, Any] = {'concentration': RealVector(Real(), 1)}

reparametrized_params: list[str] = ['concentration']

support = Sphere()

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Note this is the mean in the sense of a centroid in the submanifold that minimizes expected squared geodesic distance.

property mode

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

static infer_shapes(concentration)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters:

*args – Positional args replacing each input arg with a tuple representing the sizes of each tensor input.
**kwargs – Keywords mapping name of input arg to tuple representing the sizes of each tensor input.

Returns:

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type:

SineBivariateVonMises

Bases: Distribution

Unimodal distribution of two dependent angles on the 2-torus (\(S^1 \otimes S^1\)) given by

\[C^{-1}\exp(\kappa_1\cos(x_1-\mu_1) + \kappa_2\cos(x_2 -\mu_2) + \rho\sin(x_1 - \mu_1)\sin(x_2 - \mu_2))\]

and

\[C = (2\pi)^2 \sum_{i=0} {2i \choose i} \left(\frac{\rho^2}{4\kappa_1\kappa_2}\right)^i I_i(\kappa_1)I_i(\kappa_2),\]

where \(I_i(\cdot)\) is the modified bessel function of first kind, mu’s are the locations of the distribution, kappa’s are the concentration and rho gives the correlation between angles \(x_1\) and \(x_2\). This distribution is helpful for modeling coupled angles such as torsion angles in peptide chains.

To infer parameters, use NUTS or HMC with priors that avoid parameterizations where the distribution becomes bimodal; see note below.

Note

Sample efficiency drops as

\[\frac{\rho^2}{\kappa_1\kappa_2} \rightarrow 1\]

because the distribution becomes increasingly bimodal. To avoid inefficient sampling use the weighted_correlation parameter with a skew away from one (e.g., TransformedDistribution(Beta(5,5), AffineTransform(loc=-1, scale=2))). The weighted_correlation should be in [-1,1].

Note

The correlation and weighted_correlation params are mutually exclusive.

Note

In the context of SVI, this distribution can be used as a likelihood but not for latent variables.

Note

Normalization remains accurate for concentrations up to 10,000. Unlike Pyro, there is no assertion to verify this during initialization, as JIT-compilation would invalidate such a check.

** References: **

Probabilistic model for two dependent circular variables Singh, H., Hnizdo, V., and Demchuck, E. (2002)

Parameters:

phi_loc (np.ndarray) – location of first angle
psi_loc (np.ndarray) – location of second angle
phi_concentration (np.ndarray) – concentration of first angle
psi_concentration (np.ndarray) – concentration of second angle
correlation (np.ndarray) – correlation between the two angles
weighted_correlation (np.ndarray) – set correlation to weighted_corr * sqrt(phi_conc*psi_conc) to avoid bimodality (see note). The weighted_correlation should be in [0,1].

arg_constraints: dict[str, Any] = {'correlation': Real(), 'phi_concentration': Positive(lower_bound=0.0), 'phi_loc': Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793), 'psi_concentration': Positive(lower_bound=0.0), 'psi_loc': Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793)}

support = IndependentConstraint(Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793), 1)

max_sample_iter = 1000

norm_const() → Array[source]

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

** References: **

A New Unified Approach for the Simulation of a Wide Class of Directional Distributions John T. Kent, Asaad M. Ganeiber & Kanti V. Mardia (2018)

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Computes circular mean of distribution. Note: same as location when mapped to support [-pi, pi]

SineSkewed

Bases: Distribution

Sine-skewing [1] is a procedure for producing a distribution that breaks pointwise symmetry on a torus distribution. The new distribution is called the Sine Skewed X distribution, where X is the name of the (symmetric) base distribution. Torus distributions are distributions with support on products of circles (i.e., \(\otimes S^1\) where \(S^1 = [-pi,pi)\)). So, a 0-torus is a point, the 1-torus is a circle, and the 2-torus is commonly associated with the donut shape.

The sine skewed X distribution is parameterized by a weight parameter for each dimension of the event of X. For example with a von Mises distribution over a circle (1-torus), the sine skewed von Mises distribution has one skew parameter. The skewness parameters can be inferred using HMC or NUTS. For example, the following will produce a prior over skewness for the 2-torus,:

@numpyro.handlers.reparam(config={'phi_loc': CircularReparam(), 'psi_loc': CircularReparam()})
def model(obs):
    # Sine priors
    phi_loc = numpyro.sample('phi_loc', VonMises(pi, 2.))
    psi_loc = numpyro.sample('psi_loc', VonMises(-pi / 2, 2.))
    phi_conc = numpyro.sample('phi_conc', Beta(1., 1.))
    psi_conc = numpyro.sample('psi_conc', Beta(1., 1.))
    corr_scale = numpyro.sample('corr_scale', Beta(2., 5.))

    # Skewing prior
    ball_trans = L1BallTransform()
    skewness = numpyro.sample('skew_phi', Normal(0, 0.5).expand((2,)))
    skewness = ball_trans(skewness)  # constraint sum |skewness_i| <= 1

    with numpyro.plate('obs_plate'):
        sine = SineBivariateVonMises(phi_loc=phi_loc, psi_loc=psi_loc,
                                     phi_concentration=70 * phi_conc,
                                     psi_concentration=70 * psi_conc,
                                     weighted_correlation=corr_scale)
        return numpyro.sample('phi_psi', SineSkewed(sine, skewness), obs=obs)

To ensure the skewing does not alter the normalization constant of the (sine bivariate von Mises) base distribution the skewness parameters are constraint. The constraint requires the sum of the absolute values of skewness to be less than or equal to one. We can use the L1BallTransform to achieve this.

In the context of SVI, this distribution can freely be used as a likelihood, but use as latent variables it will lead to slow inference for 2 and higher dim toruses. This is because the base_dist cannot be reparameterized.

Note

An event in the base distribution must be on a d-torus, so the event_shape must be (d,).

Note

For the skewness parameter, it must hold that the sum of the absolute value of its weights for an event must be less than or equal to one. See eq. 2.1 in [1].

** References: **

Sine-skewed toroidal distributions and their application in protein bioinformatics
Ameijeiras-Alonso, J., Ley, C. (2019)

Parameters:

base_dist (numpyro.distributions.Distribution) – base density on a d-dimensional torus. Supported base distributions include: 1D VonMises, SineBivariateVonMises, 1D ProjectedNormal, and Uniform (-pi, pi).
skewness (jax.numpy.array) – skewness of the distribution.

arg_constraints: dict[str, Any] = {'skewness': L1Ball()}

pytree_data_fields: tuple[str, ...] = ('base_dist', 'skewness')

support = IndependentConstraint(Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793), 1)

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the base distribution

VonMises

Bases: Distribution

The von Mises distribution, also known as the circular normal distribution.

This distribution is supported by a circular constraint from -pi to +pi. By default, the circular support behaves like constraints.interval(-math.pi, math.pi). To avoid issues at the boundaries of this interval during sampling, you should reparameterize this distribution using handlers.reparam with a CircularReparam reparametrizer in the model, e.g.:

@handlers.reparam(config={"direction": CircularReparam()})
def model():
    direction = numpyro.sample("direction", VonMises(0.0, 4.0))
    ...

arg_constraints: dict[str, Any] = {'concentration': Positive(lower_bound=0.0), 'loc': Real()}

reparametrized_params: list[str] = ['loc']

support = Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793)

Generate sample from von Mises distribution

Parameters:

key – random number generator key
sample_shape – shape of samples

Returns:

samples from von Mises

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Computes circular mean of distribution. NOTE: same as location when mapped to support [-pi, pi]

property variance: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Computes circular variance of distribution

Truncated Distributions

DoublyTruncatedPowerLaw

Bases: Distribution

Power law distribution with \(\alpha\) index, and lower and upper bounds. We can define the power law distribution as,

\[f(x; \alpha, a, b) = \frac{x^{\alpha}}{Z(\alpha, a, b)},\]

where, \(a\) and \(b\) are the lower and upper bounds respectively, and \(Z(\alpha, a, b)\) is the normalization constant. It is defined as,

\[\begin{split}Z(\alpha, a, b) = \begin{cases} \log(b) - \log(a) & \text{if } \alpha = -1, \\ \frac{b^{1 + \alpha} - a^{1 + \alpha}}{1 + \alpha} & \text{otherwise}. \end{cases}\end{split}\]

Parameters:

alpha – index of the power law distribution
low – lower bound of the distribution
high – upper bound of the distribution

arg_constraints: dict[str, Any] = {'alpha': Real(), 'high': GreaterThan(lower_bound=0), 'low': GreaterThanEq(lower_bound=0)}

reparametrized_params: list[str] = ['alpha', 'low', 'high']

pytree_aux_fields: tuple[str, ...] = ('_support',)

pytree_data_fields: tuple[str, ...] = ('alpha', 'low', 'high')

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Logarithmic probability distribution:

Z inequal minus one:

\[\frac{(\alpha + 1)x^\alpha}{b^{\alpha + 1} - a^{\alpha + 1}}\]

Z equal minus one:

\[\frac{x^\alpha}{\log(b) - \log(a)}\]

Derivations are calculated by Wolfram Alpha via the Jacobian matrix accordingly.

Cumulated probability distribution: Z inequal minus one:

\[\frac{x^{\alpha + 1} - a^{\alpha + 1}}{b^{\alpha + 1} - a^{\alpha + 1}}\]

Z equal minus one:

\[\frac{\log(x) - \log(a)}{\log(b) - \log(a)}\]

Derivations are calculated by Wolfram Alpha via the Jacobian matrix accordingly.

Inverse cumulated probability distribution: Z inequal minus one:

\[a \left(\frac{b}{a}\right)^{q}\]

Z equal minus one:

\[\left(a^{1 + \alpha} + q (b^{1 + \alpha} - a^{1 + \alpha})\right)^{\frac{1}{1 + \alpha}}\]

Derivations are calculated by Wolfram Alpha via the Jacobian matrix accordingly.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

LeftTruncatedDistribution

Bases: Distribution

arg_constraints: dict[str, Any] = {'low': Real()}

reparametrized_params: list[str] = ['low']

supported_types = (<class 'numpyro.distributions.continuous.Cauchy'>, <class 'numpyro.distributions.continuous.Laplace'>, <class 'numpyro.distributions.continuous.Logistic'>, <class 'numpyro.distributions.continuous.Normal'>, <class 'numpyro.distributions.continuous.SoftLaplace'>, <class 'numpyro.distributions.continuous.StudentT'>)

pytree_data_fields: tuple[str, ...] = ('base_dist', 'low', '_support')

base_dist: Cauchy | Laplace | Logistic | Normal | SoftLaplace | StudentT

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property var: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray

LowerTruncatedPowerLaw

Bases: Distribution

Lower truncated power law distribution with \(\alpha\) index. We can define the power law distribution as,

\[f(x; \alpha, a) = (-\alpha-1)a^{-\alpha - 1}x^{-\alpha}, \qquad x \geq a, \qquad \alpha < -1,\]

where, \(a\) is the lower bound. The cdf of the distribution is given by,

\[F(x; \alpha, a) = 1 - \left(\frac{x}{a}\right)^{1+\alpha}.\]

The k-th moment of the distribution is given by,

\[\begin{split}E[X^k] = \begin{cases} \frac{-\alpha-1}{-\alpha-1-k}a^k & \text{if } k < -\alpha-1, \\ \infty & \text{otherwise}. \end{cases}\end{split}\]

Parameters:

alpha – index of the power law distribution
low – lower bound of the distribution

arg_constraints: dict[str, Any] = {'alpha': LessThan(upper_bound=-1.0), 'low': GreaterThan(lower_bound=0.0)}

reparametrized_params: list[str] = ['alpha', 'low']

pytree_aux_fields: tuple[str, ...] = ('_support',)

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

RightTruncatedDistribution

Bases: Distribution

arg_constraints: dict[str, Any] = {'high': Real()}

reparametrized_params: list[str] = ['high']

supported_types = (<class 'numpyro.distributions.continuous.Cauchy'>, <class 'numpyro.distributions.continuous.Laplace'>, <class 'numpyro.distributions.continuous.Logistic'>, <class 'numpyro.distributions.continuous.Normal'>, <class 'numpyro.distributions.continuous.SoftLaplace'>, <class 'numpyro.distributions.continuous.StudentT'>)

pytree_data_fields: tuple[str, ...] = ('base_dist', 'high', '_support')

base_dist: Cauchy | Laplace | Logistic | Normal | SoftLaplace | StudentT

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property var: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray

TruncatedCauchy

class TruncatedCauchy(loc: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray = 0.0, scale: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray = 1.0, *, low: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, high: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, validate_args: bool | None = None)[source]: Bases:

TruncatedDistribution

A function to generate a truncated distribution.

Parameters:

base_dist – The base distribution to be truncated. This should be a univariate distribution. Currently, only the following distributions are supported: Cauchy, Laplace, Logistic, Normal, and StudentT.
low – the value which is used to truncate the base distribution from below. Setting this parameter to None to not truncate from below.
high – the value which is used to truncate the base distribution from above. Setting this parameter to None to not truncate from above.

TruncatedNormal

class TruncatedNormal(loc: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray = 0.0, scale: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray = 1.0, *, low: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, high: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray | None = None, validate_args: bool | None = None)[source]: Bases:

TruncatedPolyaGamma

class TruncatedPolyaGamma(batch_shape: tuple[int, ...] = (), *, validate_args: bool | None = None)[source]

Bases: Distribution

truncation_point = 2.5

num_log_prob_terms = 7

num_gamma_variates = 8

arg_constraints: dict[str, Any] = {}

support = Interval(lower_bound=0.0, upper_bound=2.5)

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

TwoSidedTruncatedDistribution

Bases: Distribution

arg_constraints: dict[str, Any] = {'high': Dependent(), 'low': Dependent()}

reparametrized_params: list[str] = ['low', 'high']

supported_types = (<class 'numpyro.distributions.continuous.Cauchy'>, <class 'numpyro.distributions.continuous.Laplace'>, <class 'numpyro.distributions.continuous.Logistic'>, <class 'numpyro.distributions.continuous.Normal'>, <class 'numpyro.distributions.continuous.SoftLaplace'>, <class 'numpyro.distributions.continuous.StudentT'>)

pytree_data_fields: tuple[str, ...] = ('base_dist', 'low', 'high', '_support')

base_dist: Cauchy | Laplace | Logistic | Normal | SoftLaplace | StudentT

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

The inverse cumulative distribution function of this distribution.

Parameters:: q – quantile values, should belong to [0, 1].
Returns:: the samples whose cdf values equals to q.

The cumulative distribution function of this distribution.

Parameters:: value – samples from this distribution.
Returns:: output of the cumulative distribution function evaluated at value.

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

property mean: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray: Mean of the distribution.

property var: Array | ndarray | bool | number | bool | int | float | complex | TypedNdArray

Censored Distributions

LeftCensoredDistribution

Bases: Distribution

Distribution wrapper for left-censored outcomes.

This distribution augments a base distribution with left-censoring, so that the likelihood contribution depends on the censoring indicator.

Parameters:

base_dist (numpyro.distributions.Distribution) – Parametric distribution for the uncensored values (e.g., Exponential, Weibull, LogNormal, Normal, etc.). This distribution must implement a cdf method.
censored (array-like of {0,1}) – Censoring indicator per observation: 0 → value is observed exactly 1 → observation is left-censored at the reported value (true value occurred on or before the reported value)

Note

The log_prob(value) method expects value to be the observed upper bound for each observation. The contribution to the log-likelihood is:

log f(value) if censored == 0 log F(value) if censored == 1

where f is the density and F the cumulative distribution function of base_dist.

This is commonly used in survival analysis, where event times are positive, but the approach is more general and can be applied to any distribution with a cumulative distribution function, regardless of support.

In R’s survival package notation, this corresponds to Surv(time, event, type = 'left').

Example:

Surv(time = c(2, 4, 6), event = c(0, 1, 0), type=’left’)

means:

subject 1 had an event exactly at t=2 subject 2 had an event before or at t=4 (left-censored) subject 3 had an event exactly at t=6

Example:

>>> from jax import numpy as jnp
>>> from numpyro import distributions as dist
>>> base = dist.LogNormal(0., 1.)
>>> surv_dist = dist.LeftCensoredDistribution(base, censored=jnp.array([0, 1, 1]))
>>> loglik = surv_dist.log_prob(jnp.array([2., 4., 6.]))

arg_constraints: dict[str, Any] = {'censored': Boolean()}

pytree_data_fields: tuple[str, ...] = ('base_dist', 'censored', '_support')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

RightCensoredDistribution

Bases: Distribution

Distribution wrapper for right-censored outcomes.

This distribution augments a base distribution with right-censoring, so that the likelihood contribution depends on the censoring indicator.

Parameters:

base_dist (numpyro.distributions.Distribution) – Parametric distribution for the uncensored values (e.g., Exponential, Weibull, LogNormal, Normal, etc.). This distribution must implement a cdf method.
censored (array-like of {0,1}) – Censoring indicator per observation: 0 → value is observed exactly 1 → observation is right-censored at the reported value (true value occurred on or after the reported value)

Note

The log_prob(value) method expects value to be the observed lower bound for each observation. The contribution to the log-likelihood is:

log f(value) if censored == 0 log (1 - F(value)) if censored == 1

where f is the density and F the cumulative distribution function of base_dist.

This is commonly used in survival analysis, where event times are positive, but the approach is more general and can be applied to any distribution with a cumulative distribution function, regardless of support.

In R’s survival package notation, this corresponds to Surv(time, event, type = 'right').

Example:

Surv(time = c(5, 8, 10), event = c(1, 0, 1))

means:

subject 1 had an event at t=5 subject 2 was censored at t=8 subject 3 had an event at t=10

Example:

>>> from jax import numpy as jnp
>>> from numpyro import distributions as dist
>>> base = dist.Exponential(rate=0.1)
>>> surv_dist = dist.RightCensoredDistribution(base, censored=jnp.array([0, 1, 0]))
>>> loglik = surv_dist.log_prob(jnp.array([5., 8., 10.]))

arg_constraints: dict[str, Any] = {'censored': Boolean()}

pytree_data_fields: tuple[str, ...] = ('base_dist', 'censored', '_support')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type:

property support: Constraint: The support of this distribution. Subclasses can override this as a class attribute or as a property.

Evaluates the log probability density for a batch of samples given by value.

Parameters:: value – A batch of samples from the distribution.
Returns:: an array with shape value.shape[:-self.event_shape]
Return type:: ArrayLike

IntervalCensoredDistribution

Bases: Distribution

Distribution wrapper for interval-censored outcomes.

This distribution augments a base distribution with interval censoring, so that the likelihood contribution depends on whether the observation is exactly observed, left-censored, right-censored, interval-censored, or doubly-censored (i.e., known to lie outside the observed interval).

Parameters:

base_dist (numpyro.distributions.Distribution) – Parametric distribution for the uncensored values (e.g., Exponential, Weibull, LogNormal, Normal, etc.). This distribution must implement a cdf method.
left_censored (array-like of {0,1}) – Indicator per observation: 1 → observation is left-censored at the reported upper bound 0 → not left-censored
right_censored (array-like of {0,1}) – Indicator per observation: 1 → observation is right-censored at the reported lower bound 0 → not right-censored

Note

The log_prob(value) method expects value to be a two-dimensional array of shape (batch_size, 2), where each row is (lower, upper). The contribution to the log-likelihood is determined as follows:

log F(upper) if left_censored == 1 and right_censored == 0 log (1 - F(lower)) if right_censored == 1 and left_censored == 0 log (F(upper) - F(lower)) if both == 0 (interval-censored) log (1 - (F(upper) - F(lower))) if both == 1 (doubly-censored) log f(value) if lower ≈ upper (point interval)

where f is the density and F the cumulative distribution function of base_dist.

This is commonly used in survival analysis, where event times are positive, but the approach is general and can be applied to any distribution with a cumulative distribution function, regardless of support.

In R’s survival package notation, this corresponds to Surv(l, r, type = 'interval2').

Example:

Surv(l = c(2, 4, 6), r = c(5, Inf, 9), type = ‘interval2’)

means:

subject 1 had an event in (2, 5] subject 2 was right-censored at 4 subject 3 had an event in (6, 9]

Example:

>>> from jax import numpy as jnp
>>> from numpyro import distributions as dist
>>> base = dist.Weibull(concentration=2.0, scale=3.0)
>>> left_censored = jnp.array([0, 0, 0])
>>> right_censored = jnp.array([0, 1, 0])
>>> surv_dist = dist.IntervalCensoredDistribution(base, left_censored, right_censored)
>>> values = jnp.array([
...     [2.0, 5.0],
...     [4.0, jnp.inf],
...     [6.0, 9.0],
... ])
>>> loglik = surv_dist.log_prob(values)

arg_constraints: dict[str, Any] = {'left_censored': Boolean(), 'right_censored': Boolean()}

pytree_data_fields: tuple[str, ...] = ('base_dist', 'left_censored', 'right_censored', '_support')

Returns a sample from the distribution having shape given by sample_shape + batch_shape + event_shape. Note that when sample_shape is non-empty, leading dimensions (of size sample_shape) of the returned sample will be filled with iid draws from the distribution instance.

Parameters:

key (jax.random.key) – the rng_key key to be used for the distribution.
sample_shape (tuple) – the sample shape for the distribution.

Returns:

an array of shape sample_shape + batch_shape + event_shape

Return type: