tfp.substrates.jax.distributions.BatchBroadcast

A distribution that broadcasts an underlying distribution's batch shape.

Inherits From: AutoCompositeTensorDistribution, Distribution

View aliases

Main aliases

tfp.experimental.substrates.jax.distributions.BatchBroadcast

tfp.substrates.jax.distributions.BatchBroadcast(
    distribution,
    with_shape=None,
    *,
    to_shape=None,
    validate_args=False,
    name=None
)

This meta-distribution can be useful when we desire to implicitly broadcast an underlying distribution's batch shape with, or to, another shape, typically to parameterize a larger batch of distributions.

This distribution supports two flavors of broadcasting. The with_shape argument broadcasts the underlying distribution's batch shape with a compatible shape with_shape, obtaining a batch shape that results from the broadcast of these two shapes together. Alternatively, the to_shape argument broadcasts the underlying distribution's batch shape to the exact shape specified. With an unnamed argument, the more permissive with_shape behavior is used.

#### Examples

  d = tfd.BatchBroadcast(tfd.Normal(tf.range(3.), 1.), with_shape=[2, 3])
  d.batch_shape  # => [2, 3]
  d.distribution.batch_shape  # => [3]
  d.event_shape  # => []

  d = tfd.BatchBroadcast(tfd.Normal(tf.range(3.), 1.), to_shape=[2, 3])
  d.batch_shape  # => [2, 3]

  df = tfd.Uniform(4., 5.).sample([10, 1])
  d = tfd.BatchBroadcast(
      tfd.WishartTriL(df=df, scale_tril=tf.eye(3)), with_shape=[2])
  d.batch_shape  # => [10, 2]
  d.distribution.batch_shape  # => [10, 1]
  d.event_shape  # => [3, 3]

  d = tfd.BatchBroadcast(
      tfd.WishartTriL(df=df, scale_tril=tf.eye(3)), to_shape=[2])
  # => Exception: to_shape is too small

  d = tfd.BatchBroadcast(tfd.WishartTriL(df=df, scale_tril=tf.eye(3)),
                         to_shape=[10, 2])
  d.batch_shape  # => [10, 2]

#### Example: Spatially distributed samples

In some cases a particular batch shape may be required, but the underlying parameterization has a smaller representation.

Suppose data is sampled in 10 different vicinities on a globe. We might write:

  loc = tfp.random.spherical_uniform([10], 3)
  components_dist = tfd.VonMisesFisher(mean_direction=loc, concentration=50.)

Now suppose we are operating 500 different experiments, each of which samples these different vicinities in different proportions. We might hope to write:

  mixture_dist = tfd.Categorical(logits=tf.random.stateless_uniform([500, 10]))
  obs_dist = tfd.MixtureSameFamily(mixture_dist, components_dist)

But this currently (Feb. 2021) causes an exception ValueError: mixture_distribution.batch_shape ([500]) is not compatible with components_distribution.batch_shape ([]).

A naive fix would be to broadcast the parameters of components_dist to ensure the component distribution has batch shape [500, 10]. But this is wasteful in that it replicates a [10, 3]-shaped tensor 500 times, and will cause unnecessary computation, memory motion, etc. Using BatchBroadcast we may write:

  obs_dist = tfd.MixtureSameFamily(
      mixture_dist, tfd.BatchBroadcast(components_dist, [500, 10]))

This allows us to avoid avoid replicating any parameters, but achieve the requisite batch shape. If we would like to evaluate the likelihood of 20 given observation locations under each different experiment, we might write:

  test_sites = tfp.random.spherical_uniform([20], 3)
  lp = tfd.Sample(obs_dist, 20).log_prob(test_sites)  # shape [500]

If distribution is a CompositeTensor, then the resulting BatchBroadcast instance is a CompositeTensor as well. Otherwise, a non-CompositeTensor _BatchBroadcast instance is created instead. Distribution subclasses that inherit from BatchBroadcast will also inherit from CompositeTensor.

Args
distribution	The underlying distribution. Must have batch shape compatible with broadcast_shape.
with_shape	The shape with which the underlying distribution's batch shape is to be broadcast. The resulting batch shape may be different from either input. Mutually exclusive with to_shape.
to_shape	The shape to which the underlying distribution's batch shape is to be broadcast. This provides a stricter contract than with_shape, in that the resulting batch shape will be exactly the one provided in to_shape. Mutually exclusive with with_shape.
validate_args	Indicates whether additional assertions should be used; may impose a performance penalty.
name	Optional name for the distribution.

Attributes
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_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.
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	The list or structure of lists of active shard axis names.
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.
to_shape
trainable_variables
validate_args	Python bool indicating possibly expensive checks are enabled.
variables
with_shape

Methods

batch_shape_tensor

View source

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.

cdf

View source

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

View source

copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

Note: the copy distribution may continue to depend on the original initialization arguments.

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

View source

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

View source

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.

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

View source

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

Shannon entropy in nats.

event_shape_tensor

View source

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	Tensor.

experimental_default_event_space_bijector

View source

experimental_default_event_space_bijector(
    *args, **kwargs
)

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

View source

@classmethod
experimental_fit(
    value, sample_ndims=1, validate_args=False, **init_kwargs
)

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

View source

experimental_local_measure(
    value, backward_compat=False, **kwargs
)

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_sample_and_log_prob

View source

experimental_sample_and_log_prob(
    sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)

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

View source

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.

is_scalar_event

View source

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.

kl_divergence

View source

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.

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

View source

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

View source

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_survival_function

View source

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

View source

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

Mean.

mode

View source

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

Mode.

param_shapes

View source

@classmethod
param_shapes(
    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

View source

@classmethod
param_static_shapes(
    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.

parameter_properties

View source

@classmethod
parameter_properties(
    dtype=tf.float32, num_classes=None
)

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

View source

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.

quantile

View source

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

View source

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	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.

stddev

View source

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

View source

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.

unnormalized_log_prob

View source

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

Potentially unnormalized log probability density/mass function.

This function is similar to log_prob, but does not require that the return value be normalized. (Normalization here refers to the total integral of probability being one, as it should be by definition for any probability distribution.) This is useful, for example, for distributions where the normalization constant is difficult or expensive to compute. By default, this simply calls log_prob.

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

variance

View source

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().

__getitem__

View source

__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.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__

View source

__iter__()