Distributions¶

Base Distribution¶

Distribution¶

class Distribution(batch_shape=(), event_shape=(), *, validate_args=None)[source]¶

Bases: object

Base class for probability distributions in NumPyro. The design largely follows from torch.distributions.

Parameters

batch_shape – The batch shape for the distribution. This designates independent (possibly non-identical) dimensions of a sample from the distribution. This is fixed for a distribution instance and is inferred from the shape of the distribution parameters.
event_shape – The event shape for the distribution. This designates the dependent dimensions of a sample from the distribution. These are collapsed when we evaluate the log probability density of a batch of samples using .log_prob.
validate_args – Whether to enable validation of distribution parameters and arguments to .log_prob method.

As an example:

>>> import jax.numpy as jnp
>>> import numpyro.distributions as dist
>>> d = dist.Dirichlet(jnp.ones((2, 3, 4)))
>>> d.batch_shape
(2, 3)
>>> d.event_shape
(4,)

arg_constraints = {}¶

support = None¶

has_enumerate_support = False¶

reparametrized_params = []¶

pytree_data_fields = ()¶

pytree_aux_fields = ('_batch_shape', '_event_shape')¶

classmethod gather_pytree_data_fields()[source]¶

classmethod gather_pytree_aux_fields() → tuple[source]¶

tree_flatten()[source]¶

classmethod tree_unflatten(aux_data, params)[source]¶

static set_default_validate_args(value)[source]¶

property batch_shape¶

Returns the shape over which the distribution parameters are batched.

Returns: batch shape of the distribution.
Return type: tuple

property event_shape¶

Returns the shape of a single sample from the distribution without batching.

Returns: event shape of the distribution.
Return type: tuple

property event_dim¶

Returns: Number of dimensions of individual events.
Return type: int

property has_rsample¶

rsample(key, sample_shape=())[source]¶

shape(sample_shape=())[source]¶

The tensor shape of samples from this distribution.

Samples are of shape:

d.shape(sample_shape) == sample_shape + d.batch_shape + d.event_shape

Parameters: sample_shape (tuple) – the size of the iid batch to be drawn from the distribution.
Returns: shape of samples.
Return type: tuple

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

sample_with_intermediates(key, sample_shape=())[source]¶

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

Parameters

key (jax.random.PRNGKey) – 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

numpy.ndarray

log_prob(value)[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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

to_event(reinterpreted_batch_ndims=None)[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=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

expand(batch_shape)[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)[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)[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.PRNGKey(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, **kwargs)[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

tuple

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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 is_discrete¶

ExpandedDistribution¶

class ExpandedDistribution(base_dist, batch_shape=())[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {}¶

pytree_data_fields = ('base_dist',)¶

pytree_aux_fields = ('_expanded_sizes', '_interstitial_sizes')¶

property has_enumerate_support¶

bool(x) -> bool

Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.

property has_rsample¶

rsample(key, sample_shape=())[source]¶

property support¶

sample_with_intermediates(key, sample_shape=())[source]¶

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

Parameters

key (jax.random.PRNGKey) – 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

numpy.ndarray

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(value, intermediates=None)[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: numpy.ndarray

enumerate_support(expand=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

FoldedDistribution¶

class FoldedDistribution(base_dist, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.TransformedDistribution

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)¶

log_prob(*args, **kwargs)¶

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: numpy.ndarray

ImproperUniform¶

class ImproperUniform(support, batch_shape, event_shape, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {}¶

pytree_data_fields = ('support',)¶

support = Dependent()¶

log_prob(*args, **kwargs)¶

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: numpy.ndarray

Independent¶

class Independent(base_dist, reinterpreted_batch_ndims, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {}¶

pytree_data_fields = ('base_dist',)¶

pytree_aux_fields = ('reinterpreted_batch_ndims',)¶

property support¶

property has_enumerate_support¶

bool(x) -> bool

Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.

property reparametrized_params¶

Built-in mutable sequence.

If no argument is given, the constructor creates a new empty list. The argument must be an iterable if specified.

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property has_rsample¶

rsample(key, sample_shape=())[source]¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(value)[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: numpy.ndarray

expand(batch_shape)[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

MaskedDistribution¶

class MaskedDistribution(base_dist, mask)[source]¶

Bases: numpyro.distributions.distribution.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 = {}¶

pytree_data_fields = ('base_dist', '_mask')¶

pytree_aux_fields = ('_mask',)¶

property has_enumerate_support¶

bool(x) -> bool

Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.

property has_rsample¶

rsample(key, sample_shape=())[source]¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(value)[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: numpy.ndarray

enumerate_support(expand=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

tree_flatten()[source]¶

classmethod tree_unflatten(aux_data, params)[source]¶

TransformedDistribution¶

class TransformedDistribution(base_distribution, transforms, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {}¶

pytree_data_fields = ('base_dist', 'transforms')¶

property has_rsample¶

rsample(key, sample_shape=())[source]¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

sample_with_intermediates(key, sample_shape=())[source]¶

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

Parameters

key (jax.random.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

Delta¶

class Delta(v=0.0, log_density=0.0, event_dim=0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'log_density': Real(), 'v': Dependent()}¶

reparametrized_params = ['v', 'log_density']¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

Unit¶

class Unit(log_factor, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'log_factor': Real()}¶

support = Real()¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(value)[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: numpy.ndarray

Continuous Distributions¶

AsymmetricLaplace¶

class AsymmetricLaplace(loc=0.0, scale=1.0, asymmetry=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'asymmetry': Positive(lower_bound=0.0), 'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

reparametrized_params = ['loc', 'scale', 'asymmetry']¶

support = Real()¶

left_scale()[source]¶

right_scale()[source]¶

log_prob(value)[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: numpy.ndarray

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(value)[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.

AsymmetricLaplaceQuantile¶

class AsymmetricLaplaceQuantile(loc=0.0, scale=1.0, quantile=0.5, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'loc': Real(), 'quantile': OpenInterval(lower_bound=0.0, upper_bound=1.0), 'scale': Positive(lower_bound=0.0)}¶

reparametrized_params = ['loc', 'scale', 'quantile']¶

support = Real()¶

pytree_data_fields = ('loc', 'scale', 'quantile', '_ald')¶

log_prob(value)[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: numpy.ndarray

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(value)[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.

Beta¶

class Beta(concentration1, concentration0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0)}¶

reparametrized_params = ['concentration1', 'concentration0']¶

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

pytree_data_fields = ('concentration0', 'concentration1', '_dirichlet')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

BetaProportion¶

class BetaProportion(mean, concentration, *, validate_args=None)[source]¶

Bases: numpyro.distributions.continuous.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 = {'concentration': Positive(lower_bound=0.0), 'mean': OpenInterval(lower_bound=0.0, upper_bound=1.0)}¶

reparametrized_params = ['mean', 'concentration']¶

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

pytree_data_fields = ('concentration',)¶

CAR¶

class CAR(loc, correlation, conditional_precision, adj_matrix, *, is_sparse=False, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'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 = ['loc', 'correlation', 'conditional_precision', 'adj_matrix']¶

pytree_aux_fields = ('is_sparse', 'adj_matrix')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: 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

tuple

tree_flatten()[source]¶

classmethod tree_unflatten(aux_data, params)[source]¶

Cauchy¶

class Cauchy(loc=0.0, scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Real()¶

reparametrized_params = ['loc', 'scale']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

Chi2¶

class Chi2(df, *, validate_args=None)[source]¶

Bases: numpyro.distributions.continuous.Gamma

arg_constraints = {'df': Positive(lower_bound=0.0)}¶

reparametrized_params = ['df']¶

Dirichlet¶

class Dirichlet(concentration, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'concentration': IndependentConstraint(Positive(lower_bound=0.0), 1)}¶

reparametrized_params = ['concentration']¶

support = Simplex()¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: 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

tuple

EulerMaruyama¶

class EulerMaruyama(t, sde_fn, init_dist, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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

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

arg_constraints = {'t': OrderedVector()}¶

pytree_data_fields = ('t', 'init_dist')¶

pytree_aux_fields = ('sde_fn',)¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

Exponential¶

class Exponential(rate=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

reparametrized_params = ['rate']¶

arg_constraints = {'rate': Positive(lower_bound=0.0)}¶

support = Positive(lower_bound=0.0)¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

Gamma¶

class Gamma(concentration, rate=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}¶

support = Positive(lower_bound=0.0)¶

reparametrized_params = ['concentration', 'rate']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(x)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

GaussianCopula¶

class GaussianCopula(marginal_dist, correlation_matrix=None, correlation_cholesky=None, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'correlation_cholesky': CorrCholesky(), 'correlation_matrix': CorrMatrix()}¶

reparametrized_params = ['correlation_matrix', 'correlation_cholesky']¶

pytree_data_fields = ('marginal_dist', 'base_dist')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

correlation_matrix()[source]¶

correlation_cholesky()[source]¶

GaussianCopulaBeta¶

class GaussianCopulaBeta(concentration1, concentration0, correlation_matrix=None, correlation_cholesky=None, *, validate_args=False)[source]¶

Bases: numpyro.distributions.copula.GaussianCopula

arg_constraints = {'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 = ('concentration1', 'concentration0')¶

GaussianRandomWalk¶

class GaussianRandomWalk(scale=1.0, num_steps=1, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'scale': Positive(lower_bound=0.0)}¶

support = RealVector(Real(), 1)¶

reparametrized_params = ['scale']¶

pytree_aux_fields = ('num_steps',)¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

Gompertz¶

class Gompertz(concentration, rate=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}¶

support = Positive(lower_bound=0.0)¶

reparametrized_params = ['concentration', 'rate']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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¶: Mean of the distribution.

Gumbel¶

class Gumbel(loc=0.0, scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Real()¶

reparametrized_params = ['loc', 'scale']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

HalfCauchy¶

class HalfCauchy(scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

reparametrized_params = ['scale']¶

support = Positive(lower_bound=0.0)¶

arg_constraints = {'scale': Positive(lower_bound=0.0)}¶

pytree_data_fields = ('_cauchy', 'scale')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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¶: Mean of the distribution.

property variance¶: Variance of the distribution.

HalfNormal¶

class HalfNormal(scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

reparametrized_params = ['scale']¶

support = Positive(lower_bound=0.0)¶

arg_constraints = {'scale': Positive(lower_bound=0.0)}¶

pytree_data_fields = ('_normal', 'scale')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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¶: Mean of the distribution.

property variance¶: Variance of the distribution.

InverseGamma¶

class InverseGamma(concentration, rate=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.TransformedDistribution

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 = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}¶

reparametrized_params = ['concentration', 'rate']¶

support = Positive(lower_bound=0.0)¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(x)[source]¶

The cummulative distribution function of this distribution.

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

Kumaraswamy¶

class Kumaraswamy(concentration1, concentration0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0)}¶

reparametrized_params = ['concentration1', 'concentration0']¶

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

KL_KUMARASWAMY_BETA_TAYLOR_ORDER = 10¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

Laplace¶

class Laplace(loc=0.0, scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Real()¶

reparametrized_params = ['loc', 'scale']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

LKJ¶

class LKJ(dimension, concentration=1.0, sample_method='onion', *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.TransformedDistribution

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 = {'concentration': Positive(lower_bound=0.0)}¶

reparametrized_params = ['concentration']¶

support = CorrMatrix()¶

pytree_aux_fields = ('dimension', 'sample_method')¶

property mean¶: Mean of the distribution.

LKJCholesky¶

class LKJCholesky(dimension, concentration=1.0, sample_method='onion', *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 propotional 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 = {'concentration': Positive(lower_bound=0.0)}¶

reparametrized_params = ['concentration']¶

support = CorrCholesky()¶

pytree_data_fields = ('_beta', 'concentration')¶

pytree_aux_fields = ('dimension', 'sample_method')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

LogNormal¶

class LogNormal(loc=0.0, scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.TransformedDistribution

arg_constraints = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Positive(lower_bound=0.0)¶

reparametrized_params = ['loc', 'scale']¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(x)[source]¶

The cummulative distribution function of this distribution.

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

LogUniform¶

class LogUniform(low, high, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.TransformedDistribution

arg_constraints = {'high': Positive(lower_bound=0.0), 'low': Positive(lower_bound=0.0)}¶

reparametrized_params = ['low', 'high']¶

pytree_data_fields = ('low', 'high', '_support')¶

property support¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(x)[source]¶

The cummulative distribution function of this distribution.

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

Logistic¶

class Logistic(loc=0.0, scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Real()¶

reparametrized_params = ['loc', 'scale']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

LowRankMultivariateNormal¶

class LowRankMultivariateNormal(loc, cov_factor, cov_diag, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'cov_diag': IndependentConstraint(Positive(lower_bound=0.0), 1), 'cov_factor': IndependentConstraint(Real(), 2), 'loc': RealVector(Real(), 1)}¶

support = RealVector(Real(), 1)¶

reparametrized_params = ['loc', 'cov_factor', 'cov_diag']¶

pytree_data_fields = ('loc', 'cov_factor', 'cov_diag', '_capacitance_tril')¶

property mean¶: Mean of the distribution.

variance()[source]¶: Variance of the distribution.

scale_tril()[source]¶

covariance_matrix()[source]¶

precision_matrix()[source]¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

entropy()[source]¶

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

tuple

MatrixNormal¶

class MatrixNormal(loc, scale_tril_row, scale_tril_column, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 correlation matrix.
scale_tril_column (array_like) – Lower cholesky of columns correlation matrix.

References

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

arg_constraints = {'loc': RealVector(Real(), 1), 'scale_tril_column': LowerCholesky(), 'scale_tril_row': LowerCholesky()}¶

support = RealMatrix(Real(), 2)¶

reparametrized_params = ['loc', 'scale_tril_row', 'scale_tril_column']¶

property mean¶: Mean of the distribution.

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

MultivariateNormal¶

class MultivariateNormal(loc=0.0, covariance_matrix=None, precision_matrix=None, scale_tril=None, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'covariance_matrix': PositiveDefinite(), 'loc': RealVector(Real(), 1), 'precision_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}¶

support = RealVector(Real(), 1)¶

reparametrized_params = ['loc', 'covariance_matrix', 'precision_matrix', 'scale_tril']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

covariance_matrix()[source]¶

precision_matrix()[source]¶

property mean¶: Mean of the distribution.

property variance¶: 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

tuple

MultivariateStudentT¶

class MultivariateStudentT(df, loc=0.0, scale_tril=None, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'df': Positive(lower_bound=0.0), 'loc': RealVector(Real(), 1), 'scale_tril': LowerCholesky()}¶

support = RealVector(Real(), 1)¶

reparametrized_params = ['df', 'loc', 'scale_tril']¶

pytree_data_fields = ('df', 'loc', 'scale_tril', '_chi2')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

covariance_matrix()[source]¶

precision_matrix()[source]¶

property mean¶: Mean of the distribution.

property variance¶: 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

tuple

Normal¶

class Normal(loc=0.0, scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Real()¶

reparametrized_params = ['loc', 'scale']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

log_cdf(value)[source]¶

icdf(q)[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¶: Mean of the distribution.

property variance¶: Variance of the distribution.

Pareto¶

class Pareto(scale, alpha, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.TransformedDistribution

arg_constraints = {'alpha': Positive(lower_bound=0.0), 'scale': Positive(lower_bound=0.0)}¶

reparametrized_params = ['scale', 'alpha']¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

RelaxedBernoulli¶

RelaxedBernoulli(temperature, probs=None, logits=None, *, validate_args=None)[source]¶

RelaxedBernoulliLogits¶

class RelaxedBernoulliLogits(temperature, logits, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.TransformedDistribution

arg_constraints = {'logits': Real(), 'temperature': Positive(lower_bound=0.0)}¶

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

SoftLaplace¶

class SoftLaplace(loc, scale, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Real()¶

reparametrized_params = ['loc', 'scale']¶

log_prob(*args, **kwargs)¶

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: numpy.ndarray

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(value)[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¶: Mean of the distribution.

property variance¶: Variance of the distribution.

StudentT¶

class StudentT(df, loc=0.0, scale=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'df': Positive(lower_bound=0.0), 'loc': Real(), 'scale': Positive(lower_bound=0.0)}¶

support = Real()¶

reparametrized_params = ['df', 'loc', 'scale']¶

pytree_data_fields = ('df', 'loc', 'scale', '_chi2')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(q)[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.

Uniform¶

class Uniform(low=0.0, high=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'high': Dependent(), 'low': Dependent()}¶

reparametrized_params = ['low', 'high']¶

pytree_data_fields = ('low', 'high', '_support')¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(value)[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¶: Mean of the distribution.

property variance¶: Variance of the distribution.

static infer_shapes(low=(), high=())[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

tuple

Weibull¶

class Weibull(scale, concentration, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'concentration': Positive(lower_bound=0.0), 'scale': Positive(lower_bound=0.0)}¶

support = Positive(lower_bound=0.0)¶

reparametrized_params = ['scale', 'concentration']¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

Discrete Distributions¶

Bernoulli¶

Bernoulli(probs=None, logits=None, *, validate_args=None)[source]¶

BernoulliLogits¶

class BernoulliLogits(logits=None, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'logits': Real()}¶

support = Boolean()¶

has_enumerate_support = True¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

probs()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

enumerate_support(expand=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

BernoulliProbs¶

class BernoulliProbs(probs, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0)}¶

support = Boolean()¶

has_enumerate_support = True¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

logits()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

enumerate_support(expand=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

BetaBinomial¶

class BetaBinomial(concentration1, concentration0, total_count=1, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'concentration0': Positive(lower_bound=0.0), 'concentration1': Positive(lower_bound=0.0), 'total_count': IntegerNonnegative(lower_bound=0)}¶

has_enumerate_support = True¶

enumerate_support(expand=True)¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

pytree_data_fields = ('concentration1', 'concentration0', 'total_count', '_beta')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

Binomial¶

Binomial(total_count=1, probs=None, logits=None, *, validate_args=None)[source]¶

BinomialLogits¶

class BinomialLogits(logits, total_count=1, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'logits': Real(), 'total_count': IntegerNonnegative(lower_bound=0)}¶

has_enumerate_support = True¶

enumerate_support(expand=True)¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

probs()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

BinomialProbs¶

class BinomialProbs(probs, total_count=1, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0), 'total_count': IntegerNonnegative(lower_bound=0)}¶

has_enumerate_support = True¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

logits()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

enumerate_support(expand=True)[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=None)[source]¶

CategoricalLogits¶

class CategoricalLogits(logits, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'logits': RealVector(Real(), 1)}¶

has_enumerate_support = True¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

probs()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

enumerate_support(expand=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

CategoricalProbs¶

class CategoricalProbs(probs, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'probs': Simplex()}¶

has_enumerate_support = True¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

logits()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

enumerate_support(expand=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

DirichletMultinomial¶

class DirichletMultinomial(concentration, total_count=1, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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.

arg_constraints = {'concentration': IndependentConstraint(Positive(lower_bound=0.0), 1), 'total_count': IntegerNonnegative(lower_bound=0)}¶

pytree_data_fields = ('concentration', '_dirichlet')¶

pytree_aux_fields = ('total_count',)¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

static infer_shapes(concentration, total_count=())[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

tuple

DiscreteUniform¶

class DiscreteUniform(low=0, high=1, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'high': Dependent(), 'low': Dependent()}¶

has_enumerate_support = True¶

pytree_data_fields = ('low', 'high', '_support')¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

icdf(value)[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¶: Mean of the distribution.

property variance¶: Variance of the distribution.

enumerate_support(expand=True)[source]¶: Returns an array with shape len(support) x batch_shape containing all values in the support.

GammaPoisson¶

class GammaPoisson(concentration, rate=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 (beta) for the Gamma distribution.

arg_constraints = {'concentration': Positive(lower_bound=0.0), 'rate': Positive(lower_bound=0.0)}¶

support = IntegerNonnegative(lower_bound=0)¶

pytree_data_fields = ('concentration', 'rate', '_gamma')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

Geometric¶

Geometric(probs=None, logits=None, *, validate_args=None)[source]¶

GeometricLogits¶

class GeometricLogits(logits, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'logits': Real()}¶

support = IntegerNonnegative(lower_bound=0)¶

probs()[source]¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

GeometricProbs¶

class GeometricProbs(probs, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0)}¶

support = IntegerNonnegative(lower_bound=0)¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

logits()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

Multinomial¶

Multinomial(total_count=1, probs=None, logits=None, *, total_count_max=None, validate_args=None)[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, total_count=1, *, total_count_max=None, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'logits': RealVector(Real(), 1), 'total_count': IntegerNonnegative(lower_bound=0)}¶

pytree_data_fields = ('logits',)¶

pytree_aux_fields = ('total_count', 'total_count_max')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

probs()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

static infer_shapes(logits, total_count)[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

tuple

MultinomialProbs¶

class MultinomialProbs(probs, total_count=1, *, total_count_max=None, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'probs': Simplex(), 'total_count': IntegerNonnegative(lower_bound=0)}¶

pytree_data_fields = ('probs',)¶

pytree_aux_fields = ('total_count', 'total_count_max')¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

logits()[source]¶

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

property support¶

static infer_shapes(probs, total_count)[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

tuple

OrderedLogistic¶

class OrderedLogistic(predictor, cutpoints, *, validate_args=None)[source]¶

Bases: numpyro.distributions.discrete.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 = {'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

tuple

NegativeBinomial¶

NegativeBinomial(total_count, probs=None, logits=None, *, validate_args=None)[source]¶

NegativeBinomialLogits¶

class NegativeBinomialLogits(total_count, logits, *, validate_args=None)[source]¶

Bases: numpyro.distributions.conjugate.GammaPoisson

arg_constraints = {'logits': Real(), 'total_count': Positive(lower_bound=0.0)}¶

support = IntegerNonnegative(lower_bound=0)¶

log_prob(*args, **kwargs)¶

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: numpy.ndarray

NegativeBinomialProbs¶

class NegativeBinomialProbs(total_count, probs, *, validate_args=None)[source]¶

Bases: numpyro.distributions.conjugate.GammaPoisson

arg_constraints = {'probs': UnitInterval(lower_bound=0.0, upper_bound=1.0), 'total_count': Positive(lower_bound=0.0)}¶

support = IntegerNonnegative(lower_bound=0)¶

NegativeBinomial2¶

class NegativeBinomial2(mean, concentration, *, validate_args=None)[source]¶

Bases: numpyro.distributions.conjugate.GammaPoisson

Another parameterization of GammaPoisson with rate is replaced by mean.

arg_constraints = {'concentration': Positive(lower_bound=0.0), 'mean': Positive(lower_bound=0.0)}¶

support = IntegerNonnegative(lower_bound=0)¶

pytree_data_fields = ('concentration',)¶

Poisson¶

class Poisson(rate, *, is_sparse=False, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'rate': Positive(lower_bound=0.0)}¶

support = IntegerNonnegative(lower_bound=0)¶

pytree_aux_fields = ('is_sparse',)¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property variance¶: Variance of the distribution.

cdf(value)[source]¶

The cummulative distribution function of this distribution.

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

ZeroInflatedDistribution¶

ZeroInflatedDistribution(base_dist, *, gate=None, gate_logits=None, validate_args=None)[source]¶

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¶

class ZeroInflatedPoisson(gate, rate=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.discrete.ZeroInflatedProbs

A Zero Inflated Poisson distribution.

Parameters

gate (numpy.ndarray) – probability of extra zeros.
rate (numpy.ndarray) – rate of Poisson distribution.

arg_constraints = {'gate': UnitInterval(lower_bound=0.0, upper_bound=1.0), 'rate': Positive(lower_bound=0.0)}¶

support = IntegerNonnegative(lower_bound=0)¶

pytree_data_fields = ('rate',)¶

ZeroInflatedNegativeBinomial2¶

ZeroInflatedNegativeBinomial2(mean, concentration, *, gate=None, gate_logits=None, validate_args=None)[source]¶

Mixture Distributions¶

Mixture¶

Mixture(mixing_distribution, component_distributions, *, validate_args=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, component_distribution, *, validate_args=None)[source]¶

Bases: numpyro.distributions.mixtures._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.PRNGKey(42)).shape
()

pytree_data_fields = ('_mixing_distribution', '_component_distribution')¶

pytree_aux_fields = ('_mixture_size',)¶

property component_distribution¶

Return the vectorized distribution of components being mixed.

Returns: Component distribution
Return type: Distribution

property support¶

property is_discrete¶

property component_mean¶

property component_variance¶

component_cdf(samples)[source]¶

component_sample(key, sample_shape=())[source]¶

component_log_probs(value)[source]¶

MixtureGeneral¶

class MixtureGeneral(mixing_distribution, component_distributions, *, validate_args=None)[source]¶

Bases: numpyro.distributions.mixtures._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.

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.PRNGKey(42)).shape
()

pytree_data_fields = ('_mixing_distribution', '_component_distributions')¶

pytree_aux_fields = ('_mixture_size',)¶

property component_distributions¶

The list of component distributions in the mixture

Returns: The list of component distributions
Return type: list[Distribution]

property support¶

property is_discrete¶

property component_mean¶

property component_variance¶

component_cdf(samples)[source]¶

component_sample(key, sample_shape=())[source]¶

component_log_probs(value)[source]¶

Directional Distributions¶

ProjectedNormal¶

class ProjectedNormal(concentration, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'concentration': RealVector(Real(), 1)}¶

reparametrized_params = ['concentration']¶

support = Sphere()¶

property mean¶: Note this is the mean in the sense of a centroid in the submanifold that minimizes expected squared geodesic distance.

property mode¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(value)[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: numpy.ndarray

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

tuple

SineBivariateVonMises¶

class SineBivariateVonMises(phi_loc, psi_loc, phi_concentration, psi_concentration, correlation=None, weighted_correlation=None, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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

\[\frac{\rho}{\kappa_1\kappa_2} \rightarrow 1\]

because the distribution becomes increasingly bimodal. To avoid bimodality use the weighted_correlation parameter with a skew away from one (e.g., Beta(1,3)). The weighted_correlation should be in [0,1].

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.

** 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 = {'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()[source]¶

log_prob(*args, **kwargs)¶

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: numpy.ndarray

sample(key, sample_shape=())[source]¶

** 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¶: Computes circular mean of distribution. Note: same as location when mapped to support [-pi, pi]

SineSkewed¶

class SineSkewed(base_dist: numpyro.distributions.distribution.Distribution, skewness, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'skewness': L1Ball()}¶

pytree_data_fields = ('base_dist', 'skewness')¶

support = IndependentConstraint(Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793), 1)¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(value)[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: numpy.ndarray

property mean¶: Mean of the base distribution

VonMises¶

class VonMises(loc, concentration, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.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 = {'concentration': Positive(lower_bound=0.0), 'loc': Real()}¶

reparametrized_params = ['loc']¶

support = Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793)¶

sample(key, sample_shape=())[source]¶

Generate sample from von Mises distribution

Parameters

key – random number generator key
sample_shape – shape of samples

Returns

samples from von Mises

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Computes circular mean of distribution. NOTE: same as location when mapped to support [-pi, pi]

property variance¶: Computes circular variance of distribution

Truncated Distributions¶

LeftTruncatedDistribution¶

class LeftTruncatedDistribution(base_dist, low=0.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'low': Real()}¶

reparametrized_params = ['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 = ('base_dist', 'low', '_support')¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property var¶

RightTruncatedDistribution¶

class RightTruncatedDistribution(base_dist, high=0.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'high': Real()}¶

reparametrized_params = ['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 = ('base_dist', 'high', '_support')¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property var¶

TruncatedCauchy¶

class TruncatedCauchy(loc=0.0, scale=1.0, *, low=None, high=None, validate_args=None)[source]¶: Bases:

TruncatedDistribution¶

TruncatedDistribution(base_dist, low=None, high=None, *, validate_args=None)[source]¶

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=0.0, scale=1.0, *, low=None, high=None, validate_args=None)[source]¶: Bases:

TruncatedPolyaGamma¶

class TruncatedPolyaGamma(batch_shape=(), *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

truncation_point = 2.5¶

num_log_prob_terms = 7¶

num_gamma_variates = 8¶

arg_constraints = {}¶

support = Interval(lower_bound=0.0, upper_bound=2.5)¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

TwoSidedTruncatedDistribution¶

class TwoSidedTruncatedDistribution(base_dist, low=0.0, high=1.0, *, validate_args=None)[source]¶

Bases: numpyro.distributions.distribution.Distribution

arg_constraints = {'high': Dependent(), 'low': Dependent()}¶

reparametrized_params = ['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 = ('base_dist', 'low', 'high', '_support')¶

property support¶

sample(key, sample_shape=())[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.PRNGKey) – 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

numpy.ndarray

log_prob(*args, **kwargs)¶

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: numpy.ndarray

property mean¶: Mean of the distribution.

property var¶

TensorFlow Distributions¶

Thin wrappers around TensorFlow Probability (TFP) distributions. For details on the TFP distribution interface, see its Distribution docs.

BijectorConstraint¶

class BijectorConstraint(bijector)[source]¶

A constraint which is codomain of a TensorFlow bijector.

Parameters: bijector (Bijector) – a TensorFlow bijector

BijectorTransform¶

class BijectorTransform(bijector)[source]¶

A wrapper for TensorFlow bijectors to make them compatible with NumPyro’s transforms.

Parameters: bijector (Bijector) – a TensorFlow bijector

TFPDistribution¶

class TFPDistribution(batch_shape=(), event_shape=(), *, validate_args=None)[source]¶

A thin wrapper for TensorFlow Probability (TFP) distributions. The constructor has the same signature as the corresponding TFP distribution.

This class can be used to convert a TFP distribution to a NumPyro-compatible one as follows:

d = TFPDistribution[tfd.Normal](0, 1)

Note that typical use cases do not require explicitly invoking this wrapper, since NumPyro wraps TFP distributions automatically under the hood in model code, e.g.:

from tensorflow_probability.substrates.jax import distributions as tfd

def model():
    numpyro.sample("x", tfd.Normal(0, 1))

Constraints¶

Constraint¶

class Constraint[source]¶

Bases: object

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

is_discrete = False¶

event_dim = 0¶

check(value)[source]¶: Returns a byte tensor of sample_shape + batch_shape indicating whether each event in value satisfies this constraint.

feasible_like(prototype)[source]¶: Get a feasible value which has the same shape as dtype as prototype.

classmethod tree_unflatten(aux_data, params)[source]¶

boolean¶

boolean = Boolean()¶

circular¶

circular = Circular(lower_bound=-3.141592653589793, upper_bound=3.141592653589793)¶

corr_cholesky¶

corr_cholesky = CorrCholesky()¶

corr_matrix¶

corr_matrix = CorrMatrix()¶

dependent¶

dependent = Dependent()¶

Placeholder for variables whose support depends on other variables. These variables obey no simple coordinate-wise constraints.

Parameters

is_discrete (bool) – Optional value of .is_discrete in case this can be computed statically. If not provided, access to the .is_discrete attribute will raise a NotImplementedError.
event_dim (int) – Optional value of .event_dim in case this can be computed statically. If not provided, access to the .event_dim attribute will raise a NotImplementedError.

greater_than¶

greater_than(lower_bound)¶

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

integer_interval¶

integer_interval(lower_bound, upper_bound)¶

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

integer_greater_than¶

integer_greater_than(lower_bound)¶

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

interval¶

interval(lower_bound, upper_bound)¶

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

l1_ball¶

l1_ball(x)¶: Constrain to the L1 ball of any dimension.

less_than¶

less_than(upper_bound)¶

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

lower_cholesky¶

lower_cholesky = LowerCholesky()¶

multinomial¶

multinomial(upper_bound)¶

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

nonnegative_integer¶

nonnegative_integer = IntegerNonnegative(lower_bound=0)¶

ordered_vector¶

ordered_vector = OrderedVector()¶

positive¶

positive = Positive(lower_bound=0.0)¶

positive_definite¶

positive_definite = PositiveDefinite()¶

positive_integer¶

positive_integer = IntegerPositive(lower_bound=1)¶

positive_ordered_vector¶

positive_ordered_vector = PositiveOrderedVector()¶: Constrains to a positive real-valued tensor where the elements are monotonically increasing along the event_shape dimension.

real¶

real = Real()¶

real_vector¶

real_vector = RealVector(Real(), 1)¶

scaled_unit_lower_cholesky¶

scaled_unit_lower_cholesky = ScaledUnitLowerCholesky()¶

softplus_positive¶

softplus_positive = SoftplusPositive(lower_bound=0.0)¶

softplus_lower_cholesky¶

softplus_lower_cholesky = SoftplusLowerCholesky()¶

simplex¶

simplex = Simplex()¶

sphere¶

sphere = Sphere()¶: Constrain to the Euclidean sphere of any dimension.

unit_interval¶

unit_interval = UnitInterval(lower_bound=0.0, upper_bound=1.0)¶

Transforms¶

biject_to¶

biject_to(constraint)¶

Transform¶

class Transform[source]¶

Bases: object

domain = Real()¶

codomain = Real()¶

property inv¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

call_with_intermediates(x)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

classmethod tree_unflatten(aux_data, params)[source]¶

AbsTransform¶

class AbsTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

domain = Real()¶

codomain = Positive(lower_bound=0.0)¶

AffineTransform¶

class AffineTransform(loc, scale, domain=Real())[source]¶

Bases: numpyro.distributions.transforms.Transform

Note

When scale is a JAX tracer, we always assume that scale > 0 when calculating codomain.

property codomain¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

tree_flatten()[source]¶

CholeskyTransform¶

class CholeskyTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

Transform via the mapping \(y = cholesky(x)\), where x is a positive definite matrix.

domain = PositiveDefinite()¶

codomain = LowerCholesky()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

ComposeTransform¶

class ComposeTransform(parts)[source]¶

Bases: numpyro.distributions.transforms.Transform

property domain¶

property codomain¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

call_with_intermediates(x)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

tree_flatten()[source]¶

CorrCholeskyTransform¶

class CorrCholeskyTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

Transforms a uncontrained real vector \(x\) with length \(D*(D-1)/2\) into the Cholesky factor of a D-dimension correlation matrix. This Cholesky factor is a lower triangular matrix with positive diagonals and unit Euclidean norm for each row. The transform is processed as follows:

First we convert \(x\) into a lower triangular matrix with the following order:

\[\begin{split}\begin{bmatrix} 1 & 0 & 0 & 0 \\ x_0 & 1 & 0 & 0 \\ x_1 & x_2 & 1 & 0 \\ x_3 & x_4 & x_5 & 1 \end{bmatrix}\end{split}\]

2. For each row \(X_i\) of the lower triangular part, we apply a signed version of class StickBreakingTransform to transform \(X_i\) into a unit Euclidean length vector using the following steps:

Scales into the interval \((-1, 1)\) domain: \(r_i = \tanh(X_i)\).

Transforms into an unsigned domain: \(z_i = r_i^2\).

Applies \(s_i = StickBreakingTransform(z_i)\).

Transforms back into signed domain: \(y_i = (sign(r_i), 1) * \sqrt{s_i}\).

domain = RealVector(Real(), 1)¶

codomain = CorrCholesky()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

CorrMatrixCholeskyTransform¶

class CorrMatrixCholeskyTransform[source]¶

Bases: numpyro.distributions.transforms.CholeskyTransform

Transform via the mapping \(y = cholesky(x)\), where x is a correlation matrix.

domain = CorrMatrix()¶

codomain = CorrCholesky()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

ExpTransform¶

class ExpTransform(domain=Real())[source]¶

Bases: numpyro.distributions.transforms.Transform

property codomain¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

tree_flatten()[source]¶

IdentityTransform¶

class IdentityTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

L1BallTransform¶

class L1BallTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

Transforms a uncontrained real vector \(x\) into the unit L1 ball.

domain = RealVector(Real(), 1)¶

codomain = L1Ball()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

LowerCholeskyAffine¶

class LowerCholeskyAffine(loc, scale_tril)[source]¶

Bases: numpyro.distributions.transforms.Transform

Transform via the mapping \(y = loc + scale\_tril\ @\ x\).

Parameters

loc – a real vector.
scale_tril – a lower triangular matrix with positive diagonal.

Example

>>> import jax.numpy as jnp
>>> from numpyro.distributions.transforms import LowerCholeskyAffine
>>> base = jnp.ones(2)
>>> loc = jnp.zeros(2)
>>> scale_tril = jnp.array([[0.3, 0.0], [1.0, 0.5]])
>>> affine = LowerCholeskyAffine(loc=loc, scale_tril=scale_tril)
>>> affine(base)
Array([0.3, 1.5], dtype=float32)

domain = RealVector(Real(), 1)¶

codomain = RealVector(Real(), 1)¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

tree_flatten()[source]¶

LowerCholeskyTransform¶

class LowerCholeskyTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

Transform a real vector to a lower triangular cholesky factor, where the strictly lower triangular submatrix is unconstrained and the diagonal is parameterized with an exponential transform.

domain = RealVector(Real(), 1)¶

codomain = LowerCholesky()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

OrderedTransform¶

class OrderedTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

Transform a real vector to an ordered vector.

References:

Stan Reference Manual v2.20, section 10.6, Stan Development Team

Example

>>> import jax.numpy as jnp
>>> from numpyro.distributions.transforms import OrderedTransform
>>> base = jnp.ones(3)
>>> transform = OrderedTransform()
>>> assert jnp.allclose(transform(base), jnp.array([1., 3.7182817, 6.4365635]), rtol=1e-3, atol=1e-3)

domain = RealVector(Real(), 1)¶

codomain = OrderedVector()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

PermuteTransform¶

class PermuteTransform(permutation)[source]¶

Bases: numpyro.distributions.transforms.Transform

domain = RealVector(Real(), 1)¶

codomain = RealVector(Real(), 1)¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

tree_flatten()[source]¶

PowerTransform¶

class PowerTransform(exponent)[source]¶

Bases: numpyro.distributions.transforms.Transform

domain = Positive(lower_bound=0.0)¶

codomain = Positive(lower_bound=0.0)¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

tree_flatten()[source]¶

ScaledUnitLowerCholeskyTransform¶

class ScaledUnitLowerCholeskyTransform[source]¶

Bases: numpyro.distributions.transforms.LowerCholeskyTransform

Like LowerCholeskyTransform this Transform transforms a real vector to a lower triangular cholesky factor. However it does so via a decomposition

\(y = loc + unit\_scale\_tril\ @\ scale\_diag\ @\ x\).

where \(unit\_scale\_tril\) has ones along the diagonal and \(scale\_diag\) is a diagonal matrix with all positive entries that is parameterized with a softplus transform.

domain = RealVector(Real(), 1)¶

codomain = ScaledUnitLowerCholesky()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

SigmoidTransform¶

class SigmoidTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

codomain = UnitInterval(lower_bound=0.0, upper_bound=1.0)¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

SimplexToOrderedTransform¶

class SimplexToOrderedTransform(anchor_point=0.0)[source]¶

Bases: numpyro.distributions.transforms.Transform

Transform a simplex into an ordered vector (via difference in Logistic CDF between cutpoints) Used in [1] to induce a prior on latent cutpoints via transforming ordered category probabilities.

Parameters: anchor_point – Anchor point is a nuisance parameter to improve the identifiability of the transform. For simplicity, we assume it is a scalar value, but it is broadcastable x.shape[:-1]. For more details please refer to Section 2.2 in [1]

References:

Ordinal Regression Case Study, section 2.2, M. Betancourt, https://betanalpha.github.io/assets/case_studies/ordinal_regression.html

Example

>>> import jax.numpy as jnp
>>> from numpyro.distributions.transforms import SimplexToOrderedTransform
>>> base = jnp.array([0.3, 0.1, 0.4, 0.2])
>>> transform = SimplexToOrderedTransform()
>>> assert jnp.allclose(transform(base), jnp.array([-0.8472978, -0.40546507, 1.3862944]), rtol=1e-3, atol=1e-3)

domain = Simplex()¶

codomain = OrderedVector()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

tree_flatten()[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

SoftplusLowerCholeskyTransform¶

class SoftplusLowerCholeskyTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

Transform from unconstrained vector to lower-triangular matrices with nonnegative diagonal entries. This is useful for parameterizing positive definite matrices in terms of their Cholesky factorization.

domain = RealVector(Real(), 1)¶

codomain = SoftplusLowerCholesky()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

SoftplusTransform¶

class SoftplusTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

Transform from unconstrained space to positive domain via softplus \(y = \log(1 + \exp(x))\). The inverse is computed as \(x = \log(\exp(y) - 1)\).

domain = Real()¶

codomain = SoftplusPositive(lower_bound=0.0)¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

StickBreakingTransform¶

class StickBreakingTransform[source]¶

Bases: numpyro.distributions.transforms.ParameterFreeTransform

domain = RealVector(Real(), 1)¶

codomain = Simplex()¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

forward_shape(shape)[source]¶: Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)[source]¶: Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

Flows¶

InverseAutoregressiveTransform¶

class InverseAutoregressiveTransform(autoregressive_nn, log_scale_min_clip=- 5.0, log_scale_max_clip=3.0)[source]¶

Bases: numpyro.distributions.transforms.Transform

An implementation of Inverse Autoregressive Flow, using Eq (10) from Kingma et al., 2016,

\(\mathbf{y} = \mu_t + \sigma_t\odot\mathbf{x}\)

where \(\mathbf{x}\) are the inputs, \(\mathbf{y}\) are the outputs, \(\mu_t,\sigma_t\) are calculated from an autoregressive network on \(\mathbf{x}\), and \(\sigma_t>0\).

References

Improving Variational Inference with Inverse Autoregressive Flow [arXiv:1606.04934], Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, Max Welling

domain = RealVector(Real(), 1)¶

codomain = RealVector(Real(), 1)¶

call_with_intermediates(x)[source]¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

Calculates the elementwise determinant of the log jacobian.

Parameters

x (numpy.ndarray) – the input to the transform
y (numpy.ndarray) – the output of the transform

tree_flatten()[source]¶

BlockNeuralAutoregressiveTransform¶

class BlockNeuralAutoregressiveTransform(bn_arn)[source]¶

Bases: numpyro.distributions.transforms.Transform

An implementation of Block Neural Autoregressive flow.

References

Block Neural Autoregressive Flow, Nicola De Cao, Ivan Titov, Wilker Aziz

domain = RealVector(Real(), 1)¶

codomain = RealVector(Real(), 1)¶

call_with_intermediates(x)[source]¶

log_abs_det_jacobian(x, y, intermediates=None)[source]¶

Calculates the elementwise determinant of the log jacobian.

Parameters

x (numpy.ndarray) – the input to the transform
y (numpy.ndarray) – the output of the transform

tree_flatten()[source]¶