tfp.substrates.numpy.distributions.Distribution

A generic probability distribution base class.

Distribution is a base class for constructing and organizing properties (e.g., mean, variance) of random variables (e.g, Bernoulli, Gaussian).

Subclassing

Subclasses are expected to implement a leading-underscore version of the same-named function. The argument signature should be identical except for the omission of name='...'. For example, to enable log_prob(value, name='log_prob') a subclass should implement _log_prob(value).

Subclasses can append to public-level docstrings by providing docstrings for their method specializations. For example:

@distribution_util.AppendDocstring('Some other details.')
def _log_prob(self, value):
  ...

would add the string "Some other details." to the log_prob function docstring. This is implemented as a simple decorator to avoid python linter complaining about missing Args/Returns/Raises sections in the partial docstrings.

Broadcasting, batching, and shapes

All distributions support batches of independent distributions of that type. The batch shape is determined by broadcasting together the parameters.

The shape of arguments to __init__, cdf, log_cdf, prob, and log_prob reflect this broadcasting, as does the return value of sample.

sample_n_shape = [n] + batch_shape + event_shape, where sample_n_shape is the shape of the Tensor returned from sample(n), n is the number of samples, batch_shape defines how many independent distributions there are, and event_shape defines the shape of samples from each of those independent distributions. Samples are independent along the batch_shape dimensions, but not necessarily so along the event_shape dimensions (depending on the particulars of the underlying distribution).

Using the Uniform distribution as an example:

minval = 3.0
maxval = [[4.0, 6.0],
          [10.0, 12.0]]

# Broadcasting:
# This instance represents 4 Uniform distributions. Each has a lower bound at
# 3.0 as the `minval` parameter was broadcasted to match `maxval`'s shape.
u = Uniform(minval, maxval)

# `event_shape` is `TensorShape([])`.
event_shape = u.event_shape
# `event_shape_t` is a `Tensor` which will evaluate to [].
event_shape_t = u.event_shape_tensor()

# Sampling returns a sample per distribution. `samples` has shape
# [5, 2, 2], which is [n] + batch_shape + event_shape, where n=5,
# batch_shape=[2, 2], and event_shape=[].
samples = u.sample(5)

# The broadcasting holds across methods. Here we use `cdf` as an example. The
# same holds for `log_cdf` and the likelihood functions.

# `cum_prob` has shape [2, 2] as the `value` argument was broadcasted to the
# shape of the `Uniform` instance.
cum_prob_broadcast = u.cdf(4.0)

# `cum_prob`'s shape is [2, 2], one per distribution. No broadcasting
# occurred.
cum_prob_per_dist = u.cdf([[4.0, 5.0],
                           [6.0, 7.0]])

# INVALID as the `value` argument is not broadcastable to the distribution's
# shape.
cum_prob_invalid = u.cdf([4.0, 5.0, 6.0])

Shapes

There are three important concepts associated with TensorFlow Distributions shapes:

  • Event shape describes the shape of a single draw from the distribution; it may be dependent across dimensions. For scalar distributions, the event shape is []. For a 5-dimensional MultivariateNormal, the event shape is [5].
  • Batch shape describes independent, not identically distributed draws, aka a "collection" or "bunch" of distributions.
  • Sample shape describes independent, identically distributed draws of batches from the distribution family.

The event shape and the batch shape are properties of a Distribution object, whereas the sample shape is associated with a specific call to sample or log_prob.

For detailed usage examples of TensorFlow Distributions shapes, see this tutorial

Parameter values leading to undefined statistics or distributions.

Some distributions do not have well-defined statistics for all initialization parameter values. For example, the beta distribution is parameterized by positive real numbers concentration1 and concentration0, and does not have well-defined mode if concentration1 < 1 or concentration0 < 1.

The user is given the option of raising an exception or returning NaN.

a = tf.exp(tf.matmul(logits, weights_a))
b = tf.exp(tf.matmul(logits, weights_b))

# Will raise exception if ANY batch member has a < 1 or b < 1.
dist = distributions.beta(a, b, allow_nan_stats=False)
mode = dist.mode().eval()

# Will return NaN for batch members with either a < 1 or b < 1.
dist = distributions.beta(a, b, allow_nan_stats=True)  # Default behavior
mode = dist.mode().eval()

In all cases, an exception is raised if invalid parameters are passed, e.g.

# Will raise an exception if any Op is run.
negative_a = -1.0 * a  # beta distribution by definition has a > 0.
dist = distributions.beta(negative_a, b, allow_nan_stats=True)
dist.mean().eval()

dtype The type of the event samples. None implies no type-enforcement.
reparameterization_type Instance of ReparameterizationType. If tfd.FULLY_REPARAMETERIZED, then samples from the distribution are fully reparameterized, and straight-through gradients are supported. If tfd.NOT_REPARAMETERIZED, then samples from the distribution are not fully reparameterized, and straight-through gradients are either partially unsupported or are not supported at all.
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.
allow_nan_stats Python bool, default True. When True, statistics (e.g., mean, mode, variance) use the value "NaN" to indicate the result is undefined. When False, an exception is raised if one or more of the statistic's batch members are undefined.
parameters Python dict of parameters used to instantiate this Distribution.
graph_parents Python list of graph prerequisites of this Distribution.
name Python str name prefixed to Ops created by this class. Default: subclass name.

ValueError if any member of graph_parents is None or not a Tensor.

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.

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.

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

validate_args Python bool indicating possibly expensive checks are enabled.
variables

Methods

batch_shape_tensor

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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

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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

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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

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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

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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

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Shannon entropy in nats.

event_shape_tensor

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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.

is_scalar_batch

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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

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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

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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

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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

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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

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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

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Mean.

mode

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Mode.

param_shapes

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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

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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

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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

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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

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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 or tfp.util.SeedStream instance, for seeding PRNG.
name name to give to the op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
samples a Tensor with prepended dimensions sample_shape.

stddev

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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

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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

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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__

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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__

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