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tfp.distributions.JointDistributionSequential

Class JointDistributionSequential

Joint distribution parameterized by distribution-making functions.

Inherits From: JointDistribution

Defined in python/distributions/joint_distribution_sequential.py.

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.)

Examples

tfd = tfp.distributions

# 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)

# In TFP, we can write this as:
joint = tfd.JointDistributionSequential([
                 tfd.Independent(tfd.Exponential(rate=[100, 120]), 1),  # 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
])
# (Notice the 1:1 correspondence between "math" and "code".)

x = joint.sample()
# ==> A length-5 list of Tensors
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.

Note: maker arguments are provided in reverse order of the previous elements in the list. In the example, notice that m depends on n and g in this order. The order is reversed for convenience. We reverse arguments under the heuristic observation that many graphical models have chain-like dependencies which are self-evidently topologically sorted from the human cognitive process of postulating the generative process. By feeding the previous num required args in reverse order we (often) enable a simpler maker function signature. If the maker needs to depend on distribution previous to one which is not a dependency, one must use a dummy arg, to "gobble up" the unused dependency, e.g., lambda _ignore, actual_dependency: SomeDistribution(actual_dependency).

Note: unlike other non-JointDistribution distributions in tfp.distributions, JointDistribution.sample (and subclasses) return a structure of Tensors rather than a Tensor. A structure can be anything which is list-like, e.g., a list or tuple of distribution makers. Accordingly joint.batch_shape returns a list-like structure of TensorShapes for each of the distributions' batch shapes and joint.batch_shape_tensor() returns a list-like structure of Tensors for each of the distributions' event shapes. (Same with event_shape analogues.)

__init__

__init__(
    model,
    validate_args=False,
    name=None
)

Construct the JointDistributionSequential distribution.

Args:

  • 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.
  • 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.
  • name: The name for ops managed by the distribution. Default value: None (i.e., "JointDistributionSequential").

Properties

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.

Returns:

  • allow_nan_stats: Python bool.

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.

Returns:

  • batch_shape: tuple of TensorShapes representing the batch_shape for each distribution in model.

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.

Returns:

  • event_shape: tuple of TensorShapes representing the event_shape for each distribution in model.

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.

Returns:

  • reparameterization_type: ReparameterizationType of each distribution in model.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

__getitem__

__getitem__(slices)

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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.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__

__iter__()

batch_shape_tensor

batch_shape_tensor(name='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 representing batch shape of each distribution in model.

cdf

cdf(
    value,
    name='cdf',
    **kwargs
)

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

copy(**override_parameters_kwargs)

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(
    name='covariance',
    **kwargs
)

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

cross_entropy(
    other,
    name='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, JointDistributionSequential

Args:

Returns:

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

entropy

entropy(
    name='entropy',
    **kwargs
)

Shannon entropy in nats.

Additional documentation from JointDistributionSequential:

Shannon entropy in nats.

event_shape_tensor

event_shape_tensor(name='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: tuple of Tensors representing the event_shape for each distribution in model.

is_scalar_batch

is_scalar_batch(name='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

is_scalar_event(name='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

kl_divergence(
    other,
    name='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, JointDistributionSequential

Args:

Returns:

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

log_cdf

log_cdf(
    value,
    name='log_cdf',
    **kwargs
)

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_prob(
    value,
    name='log_prob',
    **kwargs
)

Log probability density/mass function.

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:

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

log_prob_parts

log_prob_parts(
    value,
    name='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(
    value,
    name='log_survival_function',
    **kwargs
)

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

mean(
    name='mean',
    **kwargs
)

Mean.

mode

mode(
    name='mode',
    **kwargs
)

Mode.

param_shapes

param_shapes(
    cls,
    sample_shape,
    name='DistributionParamShapes'
)

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_static_shapes(
    cls,
    sample_shape
)

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.

prob

prob(
    value,
    name='prob',
    **kwargs
)

Probability density/mass function.

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:

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

prob_parts

prob_parts(
    value,
    name='prob_parts'
)

Log 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(
    value,
    name='quantile',
    **kwargs
)

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.

sample

sample(
    sample_shape=(),
    seed=None,
    name='sample',
    **kwargs
)

Generate samples of the specified shape.

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: Python integer seed for RNG
  • 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

sample_distributions(
    sample_shape=(),
    seed=None,
    value=None,
    name='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: Python integer seed for generating random numbers.
  • 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".

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

stddev(
    name='stddev',
    **kwargs
)

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(
    value,
    name='survival_function',
    **kwargs
)

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.

variance

variance(
    name='variance',
    **kwargs
)

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().