## Class `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:

```
@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`

and
`sample_n`

.

`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_n(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()
```

`__init__`

```
__init__(
dtype,
reparameterization_type,
validate_args,
allow_nan_stats,
parameters=None,
graph_parents=None,
name=None
)
```

Constructs the `Distribution`

.

**This is a private method for subclass use.**

#### Args:

: The type of the event samples.`dtype`

`None`

implies no type-enforcement.: Instance of`reparameterization_type`

`ReparameterizationType`

. If`tfd.FULLY_REPARAMETERIZED`

, this`Distribution`

can be reparameterized in terms of some standard distribution with a function whose Jacobian is constant for the support of the standard distribution. If`tfd.NOT_REPARAMETERIZED`

, then no such reparameterization is available.: Python`validate_args`

`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.: Python`allow_nan_stats`

`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.: Python`parameters`

`dict`

of parameters used to instantiate this`Distribution`

.: Python`graph_parents`

`list`

of graph prerequisites of this`Distribution`

.: Python`name`

`str`

name prefixed to Ops created by this class. Default: subclass name.

#### Raises:

: if any member of graph_parents is`ValueError`

`None`

or not a`Tensor`

.

## 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:

: Python`allow_nan_stats`

`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`

`TensorShape`

, possibly unknown.

`dtype`

The `DType`

of `Tensor`

s 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`

`TensorShape`

, possibly 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`

.

#### Returns:

An instance of `ReparameterizationType`

.

`validate_args`

Python `bool`

indicating possibly expensive checks are enabled.

## Methods

`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 to give to the op`name`

#### Returns:

:`batch_shape`

`Tensor`

.

`cdf`

```
cdf(
value,
name='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`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`cdf`

`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:

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

#### Returns:

: A new instance of`distribution`

`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')
```

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:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: Floating-point`covariance`

`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)`

, (Shanon)
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.: Python`name`

`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 (Shanon) cross entropy.

`entropy`

```
entropy(name='entropy')
```

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 to give to the op`name`

#### Returns:

:`event_shape`

`Tensor`

.

`is_scalar_batch`

```
is_scalar_batch(name='is_scalar_batch')
```

Indicates that `batch_shape == []`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`is_scalar_batch`

`bool`

scalar`Tensor`

.

`is_scalar_event`

```
is_scalar_event(name='is_scalar_event')
```

Indicates that `event_shape == []`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`is_scalar_event`

`bool`

scalar`Tensor`

.

`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 (Shanon) cross entropy, and `H[.]`

denotes (Shanon) entropy.

#### Args:

:`other`

`tfp.distributions.Distribution`

instance.: Python`name`

`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_cdf(
value,
name='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`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`logcdf`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`log_prob`

```
log_prob(
value,
name='log_prob'
)
```

Log probability density/mass function.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`log_prob`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`log_survival_function`

```
log_survival_function(
value,
name='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`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

`Tensor`

of shape `sample_shape(x) + self.batch_shape`

with values of type
`self.dtype`

.

`mean`

```
mean(name='mean')
```

Mean.

`mode`

```
mode(name='mode')
```

Mode.

`param_shapes`

```
@classmethod
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 to prepend ops with.`name`

#### Returns:

`dict`

of parameter name to `Tensor`

shapes.

`param_static_shapes`

```
@classmethod
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:

: if`ValueError`

`sample_shape`

is a`TensorShape`

and is not fully defined.

`prob`

```
prob(
value,
name='prob'
)
```

Probability density/mass function.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`prob`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`quantile`

```
quantile(
value,
name='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`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`quantile`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`sample`

```
sample(
sample_shape=(),
seed=None,
name='sample'
)
```

Generate samples of the specified shape.

Note that a call to `sample()`

without arguments will generate a single
sample.

#### Args:

: 0D or 1D`sample_shape`

`int32`

`Tensor`

. Shape of the generated samples.: Python integer seed for RNG`seed`

: name to give to the op.`name`

#### Returns:

: a`samples`

`Tensor`

with prepended dimensions`sample_shape`

.

`stddev`

```
stddev(name='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:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: Floating-point`stddev`

`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'
)
```

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`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

`Tensor`

of shape `sample_shape(x) + self.batch_shape`

with values of type
`self.dtype`

.

`variance`

```
variance(name='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:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: Floating-point`variance`

`Tensor`

with shape identical to`batch_shape + event_shape`

, i.e., the same shape as`self.mean()`

.