# oryx.distributions.JointDistributionSequential

Joint distribution parameterized by distribution-making functions.

Inherits From: JointDistribution, Distribution

This distribution enables both sampling and joint probability computation from a single model specification.

A joint distribution is a collection of possibly interdependent distributions. Like tf.keras.Sequential, the JointDistributionSequential can be specified via a list of functions (each responsible for making a tfp.distributions.Distribution-like instance). Unlike tf.keras.Sequential, each function can depend on the output of all previous elements rather than only the immediately previous.

#### Mathematical Details

The JointDistributionSequential implements the chain rule of probability. That is, the probability function of a length-d vector x is,

p(x) = prod{ p(x[i] | x[:i]) : i = 0, ..., (d - 1) }

The JointDistributionSequential is parameterized by a list comprised of either:

1. tfp.distributions.Distribution-like instances or,
2. callables which return a tfp.distributions.Distribution-like instance.

Each list element implements the i-th full conditional distribution, p(x[i] | x[:i]). The "conditioned on" elements are represented by the callable's required arguments. Directly providing a Distribution-like instance is a convenience and is semantically identical a zero argument callable.

Denote the i-th callables non-default arguments as args[i]. Since the callable is the conditional manifest, 0 <= len(args[i]) <= i - 1. When len(args[i]) < i - 1, the callable only depends on a subset of the previous distributions, specifically those at indexes: range(i - 1, i - 1 - num_args[i], -1). (See "Examples" and "Discussion" for why the order is reversed.)

Name resolution: The names ofJointDistributionSequentialcomponents are defined by explicitnamearguments passed to distributions (tfd.Normal(0., 1., name='x')) and/or by the argument names in distribution-making functions (lambda x: tfd.Normal(x., 1.)). Both approaches may be used in the same distribution, as long as they are consistent; referring to a single component by multiple names will raise aValueError`. Unnamed components will be assigned a dummy name.

#### Examples

Consider the following generative model:

e ~ Exponential(rate=[100,120])
g ~ Gamma(concentration=e[0], rate=e[1])
n ~ Normal(loc=0, scale=2.)
m ~ Normal(loc=n, scale=g)
for i = 1, ..., 12:
x[i] ~ Bernoulli(logits=m)

#### We can code this as:

tfd = tfp.distributions
joint = tfd.JointDistributionSequential([
tfd.Exponential(rate=[100, 120]),           # e
lambda    e: tfd.Gamma(concentration=e[0], rate=e[1]),    # g
tfd.Normal(loc=0, scale=2.),                 # n
lambda n, g: tfd.Normal(loc=n, scale=g)                   # m
lambda    m: tfd.Sample(tfd.Bernoulli(logits=m), 12)      # x
], batch_ndims=0, use_vectorized_map=True)

Notice the 1:1 correspondence between "math" and "code".

x = joint.sample()
# ==> A length-5 list of Tensors representing a draw/realization from each
#     distribution.
joint.log_prob(x)
# ==> A scalar `Tensor` representing the total log prob under all five
#     distributions.

joint.resolve_graph()
# ==> (('e', ()),
#      ('g', ('e',)),
#      ('n', ()),
#      ('m', ('n', 'g')),
#      ('x', ('m',)))

#### Discussion

JointDistributionSequential builds each distribution in list order; list items must be either a:

1. tfd.Distribution-like instance (e.g., e and n), or a
2. Python callable (e.g., g, m, x).

Regarding #1, an object is deemed "tfd.Distribution-like" if it has a sample, log_prob, and distribution properties, e.g., batch_shape, event_shape, dtype.

Regarding #2, in addition to using a function (or lambda), supplying a TFD "class" is also permissible, this also being a "Python callable." For example, instead of writing: lambda loc, scale: tfd.Normal(loc=loc, scale=scale) one could have simply written tfd.Normal.

Notice that directly providing a tfd.Distribution-like instance means there cannot exist a (dynamic) dependency on other distributions; it is "independent" both "computationally" and "statistically." The same is self-evidently true of zero-argument callables.

A distribution instance depends on other distribution instances through the distribution making function's required arguments. If the distribution maker has k required arguments then the JointDistributionSequential calls the maker with samples produced by the previous k distributions.

#### Vectorized sampling and model evaluation

When a joint distribution's sample method is called with a sample_shape (or the log_prob method is called on an input with multiple sample dimensions) the model must be equipped to handle additional batch dimensions. This may be done manually, or automatically by passing use_vectorized_map=True. Manual vectorization has historically been the default, but we now recommend that most users enable automatic vectorization unless they are affected by a specific issue; some known issues are listed below.

When using manually-vectorized joint distributions, each operation in the model must account for the possibility of batch dimensions in Distributions and their samples. By contrast, auto-vectorized models need only describe a single sample from the joint distribution; any batch evaluation is automated as required using tf.vectorized_map (vmap in JAX). In many cases this allows for significant simplications. For example, the following manually-vectorized tfd.JointDistributionSequential model:

model = tfd.JointDistributionSequential([
tfd.Normal(0., tf.ones([3])),
tfd.Normal(0., 1.),
lambda y, x: tfd.Normal(x[..., :2] + y[..., tf.newaxis], 1.)
])

can be written in auto-vectorized form as

model = tfd.JointDistributionSequential([
tfd.Normal(0., tf.ones([3])),
tfd.Normal(0., 1.),
lambda y, x: tfd.Normal(x[:2] + y, 1.)
],
use_vectorized_map=True)

in which we were able to avoid explicitly accounting for batch dimensions when indexing and slicing computed quantities in the third line.

Known limitations of automatic vectorization:

• A small fraction of TensorFlow ops are unsupported; models that use an unsupported op will raise an error and must be manually vectorized.
• Sampling large batches may be slow under automatic vectorization because TensorFlow's stateless samplers are currently converted using a non-vectorized while_loop. This limitation applies only in TensorFlow; vectorized samplers in JAX should be approximately as fast as manually vectorized code.
• Calling sample_distributions with nontrivial sample_shape will raise an error if the model contains any distributions that are not registered as CompositeTensors (TFP's basic distributions are usually fine, but support for wrapper distributions like tfd.Sample is a work in progress).

#### Batch semantics and (log-)densities

tl;dr: pass batch_ndims=0 unless you have a good reason not to.

Joint distributions now support 'auto-batching' semantics, in which the distribution's batch shape is derived by broadcasting the leftmost batch_ndims dimensions of its components' batch shapes. All remaining dimensions are considered to form a single 'event' of the joint distribution. If batch_ndims==0, then the joint distribution has batch shape [], and all component dimensions are treated as event shape. For example, the model

jd = tfd.JointDistributionSequential([
tfd.Normal(0., tf.ones([3])),
lambda x: tfd.Normal(x[..., tf.newaxis], tf.ones([3, 2]))
],
batch_ndims=0)

creates a joint distribution with batch shape [] and event shape ([3], [3, 2]). The log-density of a sample always has shape batch_shape, so this guarantees that jd.log_prob(jd.sample()) will evaluate to a scalar value. We could alternately construct a joint distribution with batch shape [3] and event shape ([], [2]) by setting batch_ndims=1, in which case jd.log_prob(jd.sample()) would evaluate to a value of shape [3].

Setting batch_ndims=None recovers the 'classic' batch semantics (currently still the default for backwards-compatibility reasons), in which the joint distribution's log_prob is computed by naively summing log densities from the component distributions. Since these component densities have shapes equal to the batch shapes of the individual components, to avoid broadcasting errors it is usually necessary to construct the components with identical batch shapes. For example, the component distributions in the model above have batch shapes of [3] and [3, 2] respectively, which would raise an error if summed directly, but can be aligned by wrapping with tfd.Independent, as in this model:

jd = tfd.JointDistributionSequential([
tfd.Normal(0., tf.ones([3])),
lambda x: tfd.Independent(tfd.Normal(x[..., tf.newaxis], tf.ones([3, 2])),
reinterpreted_batch_ndims=1)
],
batch_ndims=None)

Here the components both have batch shape [3], so jd.log_prob(jd.sample()) returns a value of shape [3], just as in the batch_ndims=1 case above. In fact, auto-batching semantics are equivalent to implicitly wrapping each component dist as tfd.Independent(dist, reinterpreted_batch_ndim=(dist.batch_shape.ndims - jd.batch_ndims)); the only vestigial difference is that under auto-batching semantics, the joint distribution has a single batch shape [3], while under the classic semantics the value of jd.batch_shape is a structure of the component batch shapes ([3], [3]). Such structured batch shapes will be deprecated in the future, since they are inconsistent with the definition of batch shapes used elsewhere in TFP.

#### References

[1] Dan Piponi, Dave Moore, and Joshua V. Dillon. Joint distributions for TensorFlow Probability. arXiv preprint arXiv:2001.11819_, 2020. https://arxiv.org/abs/2001.11819

model Python list of either tfd.Distribution instances and/or lambda functions which take the k previous distributions and returns a new tfd.Distribution instance.
batch_ndims int Tensor number of batch dimensions. The batch_shapes of all component distributions must be such that the prefixes of length batch_ndims broadcast to a consistent joint batch shape. Default value: None.
use_vectorized_map Python bool. Whether to use tf.vectorized_map to automatically vectorize evaluation of the model. This allows the model specification to focus on drawing a single sample, which is often simpler, but some ops may not be supported. Default value: False.
validate_args Python bool. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed. Default value: False.
experimental_use_kahan_sum Python bool. When True, we use Kahan summation to aggregate independent underlying log_prob values, which improves against the precision of a naive float32 sum. This can be noticeable in particular for large dimensions in float32. See CPU caveat on tfp.math.reduce_kahan_sum. This argument has no effect if batch_ndims is None. Default value: False.
name The name for ops managed by the distribution. Default value: None (i.e., "JointDistributionSequential").

allow_nan_stats Python bool describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

batch_ndims

batch_shape Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

dtype The DType of Tensors handled by this Distribution.
event_shape Shape of a single sample from a single batch as a TensorShape.

May be partially defined or unknown.

experimental_shard_axis_names Indicates whether part distributions have active shard axis names.
model

name Name prepended to all ops created by this Distribution.
parameters Dictionary of parameters used to instantiate this Distribution.
reparameterization_type Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

trainable_variables

use_vectorized_map

validate_args Python bool indicating possibly expensive checks are enabled.
variables

class Root

## Methods

### batch_shape_tensor

Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args
name name to give to the op

Returns
batch_shape Tensor.

### cdf

Cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

cdf(x) := P[X <= x]

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
cdf a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### copy

Creates a deep copy of the distribution.

Args
**override_parameters_kwargs String/value dictionary of initialization arguments to override with new values.

Returns
distribution A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

### covariance

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-k, vector-valued distribution, it is calculated as,

Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]

where Cov is a (batch of) k x k matrix, 0 <= (i, j) < k, and E denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), Covariance shall return a (batch of) matrices under some vectorization of the events, i.e.,

Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]

where Cov is a (batch of) k' x k' matrices, 0 <= (i, j) < k' = reduce_prod(event_shape), and Vec is some function mapping indices of this distribution's event dimensions to indices of a length-k' vector.

Args
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
covariance Floating-point Tensor with shape [B1, ..., Bn, k', k'] where the first n dimensions are batch coordinates and k' = reduce_prod(self.event_shape).

### cross_entropy

Computes the (Shannon) cross entropy.

Denote this distribution (self) by P and the other distribution by Q. Assuming P, Q are absolutely continuous with respect to one another and permit densities p(x) dr(x) and q(x) dr(x), (Shannon) cross entropy is defined as:

H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)

where F denotes the support of the random variable X ~ P.

other types with built-in registrations: JointDistributionNamed, JointDistributionNamedAutoBatched, JointDistributionSequential, JointDistributionSequentialAutoBatched

Args
other tfp.distributions.Distribution instance.
name Python str prepended to names of ops created by this function.

Returns
cross_entropy self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shannon) cross entropy.

### entropy

Shannon entropy in nats.

Shannon entropy in nats.

### event_shape_tensor

Shape of a single sample from a single batch as a 1-D int32 Tensor.

Args
name name to give to the op

Returns
event_shape Tensor.

### experimental_default_event_space_bijector

Bijector mapping the reals (R**n) to the event space of the distribution.

Distributions with continuous support may implement _default_event_space_bijector which returns a subclass of tfp.bijectors.Bijector that maps R**n to the distribution's event space. For example, the default bijector for the Beta distribution is tfp.bijectors.Sigmoid(), which maps the real line to [0, 1], the support of the Beta distribution. The default bijector for the CholeskyLKJ distribution is tfp.bijectors.CorrelationCholesky, which maps R^(k * (k-1) // 2) to the submanifold of k x k lower triangular matrices with ones along the diagonal.

The purpose of experimental_default_event_space_bijector is to enable gradient descent in an unconstrained space for Variational Inference and Hamiltonian Monte Carlo methods. Some effort has been made to choose bijectors such that the tails of the distribution in the unconstrained space are between Gaussian and Exponential.

For distributions with discrete event space, or for which TFP currently lacks a suitable bijector, this function returns None.

Args
*args Passed to implementation _default_event_space_bijector.
**kwargs Passed to implementation _default_event_space_bijector.

Returns
event_space_bijector Bijector instance or None.

### experimental_fit

Instantiates a distribution that maximizes the likelihood of x.

Args
value a Tensor valid sample from this distribution family.
sample_ndims Positive int Tensor number of leftmost dimensions of value that index i.i.d. samples. Default value: 1.
validate_args Python bool, default False. When True, distribution parameters are checked for validity despite possibly degrading runtime performance. When False, invalid inputs may silently render incorrect outputs. Default value: False.
**init_kwargs Additional keyword arguments passed through to cls.__init__. These take precedence in case of collision with the fitted parameters; for example, tfd.Normal.experimental_fit([1., 1.], scale=20.) returns a Normal distribution with scale=20. rather than the maximum likelihood parameter scale=0..

Returns
maximum_likelihood_instance instance of cls with parameters that maximize the likelihood of value.

### experimental_local_measure

Returns a log probability density together with a TangentSpace.

A TangentSpace allows us to calculate the correct push-forward density when we apply a transformation to a Distribution on a strict submanifold of R^n (typically via a Bijector in the TransformedDistribution subclass). The density correction uses the basis of the tangent space.

Args
value float or double Tensor.
backward_compat bool specifying whether to fall back to returning FullSpace as the tangent space, and representing R^n with the standard basis.
**kwargs Named arguments forwarded to subclass implementation.

Returns
log_prob a Tensor representing the log probability density, of shape sample_shape(x) + self.batch_shape with values of type self.dtype.
tangent_space a TangentSpace object (by default FullSpace) representing the tangent space to the manifold at value.

Raises
UnspecifiedTangentSpaceError if backward_compat is False and the _experimental_tangent_space attribute has not been defined.

### experimental_pin

Pins some parts, returning an unnormalized distribution object.

The calling convention is much like other JointDistribution methods (e.g. log_prob), but with the difference that not all parts are required. In this respect, the behavior is similar to that of the sample function's value argument.

### Examples:

# Given the following joint distribution:
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1., name='z'),
tfd.Normal(0., 1., name='y'),
lambda y, z: tfd.Normal(y + z, 1., name='x')
], validate_args=True)

# The following calls are all permissible and produce
# `JointDistributionPinned` objects behaving identically.
PartialXY = collections.namedtuple('PartialXY', 'x,y')
PartialX = collections.namedtuple('PartialX', 'x')
assert (jd.experimental_pin(x=2.).pins ==
jd.experimental_pin(x=2., z=None).pins ==
jd.experimental_pin(dict(x=2.)).pins ==
jd.experimental_pin(dict(x=2., y=None)).pins ==
jd.experimental_pin(PartialXY(x=2., y=None)).pins ==
jd.experimental_pin(PartialX(x=2.)).pins ==
jd.experimental_pin(None, None, 2.).pins ==
jd.experimental_pin([None, None, 2.]).pins)

Args
*args Positional arguments: a value structure or component values (see above).
**kwargs Keyword arguments: a value structure or component values (see above). May also include name, specifying a Python string name for ops generated by this method.

Returns
pinned a tfp.experimental.distributions.JointDistributionPinned with the given values pinned.

### experimental_sample_and_log_prob

Samples from this distribution and returns the log density of the sample.

The default implementation simply calls sample and log_prob:

def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)

However, some subclasses may provide more efficient and/or numerically stable implementations.

Args
sample_shape integer Tensor desired shape of samples to draw. Default value: ().
seed PRNG seed; see tfp.random.sanitize_seed for details. Default value: None.
name name to give to the op. Default value: 'sample_and_log_prob'.
**kwargs Named arguments forwarded to subclass implementation.

Returns
samples a Tensor, or structure of Tensors, with prepended dimensions sample_shape.
log_prob a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### is_scalar_batch

Indicates that batch_shape == [].

Args
name Python str prepended to names of ops created by this function.

Returns
is_scalar_batch bool scalar Tensor for each distribution in model.

### is_scalar_event

Indicates that event_shape == [].

Args
name Python str prepended to names of ops created by this function.

Returns
is_scalar_event bool scalar Tensor for each distribution in model.

### kl_divergence

Computes the Kullback--Leibler divergence.

Denote this distribution (self) by p and the other distribution by q. Assuming p, q are absolutely continuous with respect to reference measure r, the KL divergence is defined as:

KL[p, q] = E_p[log(p(X)/q(X))]
= -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q] - H[p]

where F denotes the support of the random variable X ~ p, H[., .] denotes (Shannon) cross entropy, and H[.] denotes (Shannon) entropy.

other types with built-in registrations: JointDistributionNamed, JointDistributionNamedAutoBatched, JointDistributionSequential, JointDistributionSequentialAutoBatched

Args
other tfp.distributions.Distribution instance.
name Python str prepended to names of ops created by this function.

Returns
kl_divergence self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

### log_cdf

Log cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

log_cdf(x) := Log[ P[X <= x] ]

Often, a numerical approximation can be used for log_cdf(x) that yields a more accurate answer than simply taking the logarithm of the cdf when x << -1.

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
logcdf a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### log_prob

Log probability density/mass function.

The measure methods of JointDistribution (log_prob, prob, etc.) can be called either by passing a single structure of tensors or by using named args for each part of the joint distribution state. For example,

jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.log_prob(sample) ==
jd.log_prob(value=sample) ==
jd.log_prob(z, x) ==
jd.log_prob(z=z, x=x) ==
jd.log_prob(z, x=x))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.log_prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.log_prob(**sample)

JointDistribution component distributions names are resolved via jd._flat_resolve_names(), which is implemented by each JointDistribution subclass (see subclass documentation for details). Generally, for components where a name was provided--- either explicitly as the name argument to a distribution or as a key in a dict-valued JointDistribution, or implicitly, e.g., by the argument name of a JointDistributionSequential distribution-making function---the provided name will be used. Otherwise the component will receive a dummy name; these may change without warning and should not be relied upon.

trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype  # => [tf.float32]
trivial_jd.log_prob([4.])
# ==> Tensor with shape `[]`.
lp = trivial_jd.log_prob(4.)
# ==> Tensor with shape `[]`.

Notice that in the first call, [4.] is interpreted as a list of one scalar while in the second call the input is a scalar. Hence both inputs result in identical scalar outputs. If we wanted to pass an explicit vector to the Exponential component---creating a vector-shaped batch of log_probs---we could instead write trivial_jd.log_prob(np.array([4])).

Args
*args Positional arguments: a value structure or component values (see above).
**kwargs Keyword arguments: a value structure or component values (see above). May also include name, specifying a Python string name for ops generated by this method.

Returns
log_prob a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### log_prob_parts

Log probability density/mass function.

Args
value list of Tensors in distribution_fn order for which we compute the log_prob_parts and to parameterize other ("downstream") distributions.
name name prepended to ops created by this function. Default value: "log_prob_parts".

Returns
log_prob_parts a tuple of Tensors representing the log_prob for each distribution_fn evaluated at each corresponding value.

### log_survival_function

Log survival function.

Given random variable X, the survival function is defined:

log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1 - P[X <= x] ]
= Log[ 1 - cdf(x) ]

Typically, different numerical approximations can be used for the log survival function, which are more accurate than 1 - cdf(x) when x >> 1.

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

Mean.

Mode.

### param_shapes

Shapes of parameters given the desired shape of a call to sample().

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample().

Subclasses should override class method _param_shapes.

Args
sample_shape Tensor or python list/tuple. Desired shape of a call to sample().
name name to prepend ops with.

Returns
dict of parameter name to Tensor shapes.

### param_static_shapes

param_shapes with static (i.e. TensorShape) shapes.

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample(). Assumes that the sample's shape is known statically.

Subclasses should override class method _param_shapes to return constant-valued tensors when constant values are fed.

Args
sample_shape TensorShape or python list/tuple. Desired shape of a call to sample().

Returns
dict of parameter name to TensorShape.

Raises
ValueError if sample_shape is a TensorShape and is not fully defined.

### parameter_properties

Returns a dict mapping constructor arg names to property annotations.

This dict should include an entry for each of the distribution's Tensor-valued constructor arguments.

Distribution subclasses are not required to implement _parameter_properties, so this method may raise NotImplementedError. Providing a _parameter_properties implementation enables several advanced features, including:

• Distribution batch slicing (sliced_distribution = distribution[i:j]).
• Automatic inference of _batch_shape and _batch_shape_tensor, which must otherwise be computed explicitly.
• Automatic instantiation of the distribution within TFP's internal property tests.
• Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.

In the future, parameter property annotations may enable additional functionality; for example, returning Distribution instances from tf.vectorized_map.

Args
dtype Optional float dtype to assume for continuous-valued parameters. Some constraining bijectors require advance knowledge of the dtype because certain constants (e.g., tfb.Softplus.low) must be instantiated with the same dtype as the values to be transformed.
num_classes Optional int Tensor number of classes to assume when inferring the shape of parameters for categorical-like distributions. Otherwise ignored.

Returns
parameter_properties A str ->tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

Raises
NotImplementedError if the distribution class does not implement _parameter_properties.

### prob

Probability density/mass function.

The measure methods of JointDistribution (log_prob, prob, etc.) can be called either by passing a single structure of tensors or by using named args for each part of the joint distribution state. For example,

jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.prob(sample) ==
jd.prob(value=sample) ==
jd.prob(z, x) ==
jd.prob(z=z, x=x) ==
jd.prob(z, x=x))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.prob(**sample)

JointDistribution component distributions names are resolved via jd._flat_resolve_names(), which is implemented by each JointDistribution subclass (see subclass documentation for details). Generally, for components where a name was provided--- either explicitly as the name argument to a distribution or as a key in a dict-valued JointDistribution, or implicitly, e.g., by the argument name of a JointDistributionSequential distribution-making function---the provided name will be used. Otherwise the component will receive a dummy name; these may change without warning and should not be relied upon.

trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype  # => [tf.float32]
trivial_jd.prob([4.])
# ==> Tensor with shape `[]`.
prob = trivial_jd.prob(4.)
# ==> Tensor with shape `[]`.

Notice that in the first call, [4.] is interpreted as a list of one scalar while in the second call the input is a scalar. Hence both inputs result in identical scalar outputs. If we wanted to pass an explicit vector to the Exponential component---creating a vector-shaped batch of probs---we could instead write trivial_jd.prob(np.array([4])).

Args
*args Positional arguments: a value structure or component values (see above).
**kwargs Keyword arguments: a value structure or component values (see above). May also include name, specifying a Python string name for ops generated by this method.

Returns
prob a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### prob_parts

Probability density/mass function.

Args
value list of Tensors in distribution_fn order for which we compute the prob_parts and to parameterize other ("downstream") distributions.
name name prepended to ops created by this function. Default value: "prob_parts".

Returns
prob_parts a tuple of Tensors representing the prob for each distribution_fn evaluated at each corresponding value.

### quantile

Quantile function. Aka 'inverse cdf' or 'percent point function'.

Given random variable X and p in [0, 1], the quantile is:

quantile(p) := x such that P[X <= x] == p

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
quantile a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### resolve_graph

Creates a tuple of tuples of dependencies.

This function is experimental. That said, we encourage its use and ask that you report problems to tfprobability@tensorflow.org.

Args
distribution_names list of str or None names corresponding to each of model elements. (Nones are expanding into the appropriate str.)
leaf_name str used when no maker depends on a particular model element.

Returns
graph tuple of (str tuple) pairs representing the name of each distribution (maker) and the names of its dependencies.

#### Example

d = tfd.JointDistributionSequential([
tfd.Independent(tfd.Exponential(rate=[100, 120]), 1),
lambda    e: tfd.Gamma(concentration=e[..., 0], rate=e[..., 1]),
tfd.Normal(loc=0, scale=2.),
lambda n, g: tfd.Normal(loc=n, scale=g),
])
d.resolve_graph()
# ==> (
#       ('e', ()),
#       ('g', ('e',)),
#       ('n', ()),
#       ('x', ('n', 'g')),
#     )

### sample

Generate samples of the specified shape.

Note that a call to sample() without arguments will generate a single sample.

##### kwargs:
• value: Tensors structured like type(model) used to parameterize other dependent ("downstream") distribution-making functions. Using None for any element will trigger a sample from the corresponding distribution. Default value: None (i.e., draw a sample from each distribution).

Args
sample_shape 0D or 1D int32 Tensor. Shape of the generated samples.
seed PRNG seed; see tfp.random.sanitize_seed for details.
name name to give to the op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
samples a Tensor with prepended dimensions sample_shape.

### sample_distributions

Generate samples and the (random) distributions.

Note that a call to sample() without arguments will generate a single sample.

Args
sample_shape 0D or 1D int32 Tensor. Shape of the generated samples.
seed PRNG seed; see tfp.random.sanitize_seed for details.
value list of Tensors in distribution_fn order to use to parameterize other ("downstream") distribution makers. Default value: None (i.e., draw a sample from each distribution).
name name prepended to ops created by this function. Default value: "sample_distributions".
**kwargs This is an alternative to passing a value, and achieves the same effect. Named arguments will be used to parameterize other dependent ("downstream") distribution-making functions. If a value argument is also provided, raises a ValueError.

Returns
distributions a tuple of Distribution instances for each of distribution_fn.
samples a tuple of Tensors with prepended dimensions sample_shape for each of distribution_fn.

### stddev

Standard deviation.

Standard deviation is defined as,

stddev = E[(X - E[X])**2]**0.5

where X is the random variable associated with this distribution, E denotes expectation, and stddev.shape = batch_shape + event_shape.

Args
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
stddev Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

### survival_function

Survival function.

Given random variable X, the survival function is defined:

survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### unnormalized_log_prob

Unnormalized log probability density/mass function.

The measure methods of JointDistribution (log_prob, prob, etc.) can be called either by passing a single structure of tensors or by using named args for each part of the joint distribution state. For example,

jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.unnormalized_log_prob(sample) ==
jd.unnormalized_log_prob(value=sample) ==
jd.unnormalized_log_prob(z, x) ==
jd.unnormalized_log_prob(z=z, x=x) ==
jd.unnormalized_log_prob(z, x=x))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.unnormalized_log_prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.unnormalized_log_prob(**sample)

JointDistribution component distributions names are resolved via jd._flat_resolve_names(), which is implemented by each JointDistribution subclass (see subclass documentation for details). Generally, for components where a name was provided--- either explicitly as the name argument to a distribution or as a key in a dict-valued JointDistribution, or implicitly, e.g., by the argument name of a JointDistributionSequential distribution-making function---the provided name will be used. Otherwise the component will receive a dummy name; these may change without warning and should not be relied upon.

trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype  # => [tf.float32]
trivial_jd.unnormalized_log_prob([4.])
# ==> Tensor with shape `[]`.
lp = trivial_jd.unnormalized_log_prob(4.)
# ==> Tensor with shape `[]`.

Notice that in the first call, [4.] is interpreted as a list of one scalar while in the second call the input is a scalar. Hence both inputs result in identical scalar outputs. If we wanted to pass an explicit vector to the Exponential component---creating a vector-shaped batch of unnormalized_log_probs---we could instead write trivial_jd.unnormalized_log_prob(np.array([4])).

Args
*args Positional arguments: a value structure or component values (see above).
**kwargs Keyword arguments: a value structure or component values (see above). May also include name, specifying a Python string name for ops generated by this method.

Returns
log_prob a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

### unnormalized_log_prob_parts

Unnormalized log probability density/mass function.

Args
value list of Tensors in distribution_fn order for which we compute the unnormalized_log_prob_parts and to parameterize other ("downstream") distributions.
name name prepended to ops created by this function. Default value: "unnormalized_log_prob_parts".

Returns
unnormalized_log_prob_parts a tuple of Tensors representing the unnormalized_log_prob for each distribution_fn evaluated at each corresponding value.

### unnormalized_prob_parts

Unnormalized probability density/mass function.

Args
value list of Tensors in distribution_fn order for which we compute the unnormalized_prob_parts and to parameterize other ("downstream") distributions.
name name prepended to ops created by this function. Default value: "unnormalized_prob_parts".

Returns
unnormalized_prob_parts a tuple of Tensors representing the unnormalized_prob for each distribution_fn evaluated at each corresponding value.

### variance

Variance.

Variance is defined as,

Var = E[(X - E[X])**2]

where X is the random variable associated with this distribution, E denotes expectation, and Var.shape = batch_shape + event_shape.

Args
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
variance Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

### __getitem__

Slices the batch axes of this distribution, returning a new instance.

b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape  # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape  # => [3, 1, 5, 2, 4]

x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape  # => [4, 5, 3]
mvn.event_shape  # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => [2]

Args
slices slices from the [] operator

Returns
dist A new tfd.Distribution instance with sliced parameters.

### __iter__

[{ "type": "thumb-down", "id": "missingTheInformationINeed", "label":"Missing the information I need" },{ "type": "thumb-down", "id": "tooComplicatedTooManySteps", "label":"Too complicated / too many steps" },{ "type": "thumb-down", "id": "outOfDate", "label":"Out of date" },{ "type": "thumb-down", "id": "samplesCodeIssue", "label":"Samples / code issue" },{ "type": "thumb-down", "id": "otherDown", "label":"Other" }]
[{ "type": "thumb-up", "id": "easyToUnderstand", "label":"Easy to understand" },{ "type": "thumb-up", "id": "solvedMyProblem", "label":"Solved my problem" },{ "type": "thumb-up", "id": "otherUp", "label":"Other" }]