tf.contrib.distributions.TransformedDistribution

A Transformed Distribution.

Inherits From: Distribution

tf.contrib.distributions.TransformedDistribution(
    distribution, bijector=None, batch_shape=None, event_shape=None,
    validate_args=False, name=None
)

A TransformedDistribution models p(y) given a base distribution p(x), and a deterministic, invertible, differentiable transform, Y = g(X). The transform is typically an instance of the Bijector class and the base distribution is typically an instance of the Distribution class.

A Bijector is expected to implement the following functions:

forward,
inverse,
inverse_log_det_jacobian. The semantics of these functions are outlined in the Bijector documentation.

We now describe how a TransformedDistribution alters the input/outputs of a Distribution associated with a random variable (rv) X.

Write cdf(Y=y) for an absolutely continuous cumulative distribution function of random variable Y; write the probability density function pdf(Y=y) := d^k / (dy_1,...,dy_k) cdf(Y=y) for its derivative wrt to Y evaluated at y. Assume that Y = g(X) where g is a deterministic diffeomorphism, i.e., a non-random, continuous, differentiable, and invertible function. Write the inverse of g as X = g^{-1}(Y) and (J o g)(x) for the Jacobian of g evaluated at x.

A TransformedDistribution implements the following operations:

sample Mathematically: Y = g(X) Programmatically: bijector.forward(distribution.sample(...))
log_prob Mathematically: `(log o pdf)(Y=y) = (log o pdf o g^{-1})(y)
```
               + (log o abs o det o J o g^{-1})(y)`
```
Programmatically: (distribution.log_prob(bijector.inverse(y)) + bijector.inverse_log_det_jacobian(y))
log_cdf Mathematically: (log o cdf)(Y=y) = (log o cdf o g^{-1})(y) Programmatically: distribution.log_cdf(bijector.inverse(x))
and similarly for: cdf, prob, log_survival_function, survival_function.

A simple example constructing a Log-Normal distribution from a Normal distribution:

ds = tfp.distributions
log_normal = ds.TransformedDistribution(
  distribution=ds.Normal(loc=0., scale=1.),
  bijector=ds.bijectors.Exp(),
  name="LogNormalTransformedDistribution")

A LogNormal made from callables:

ds = tfp.distributions
log_normal = ds.TransformedDistribution(
  distribution=ds.Normal(loc=0., scale=1.),
  bijector=ds.bijectors.Inline(
    forward_fn=tf.exp,
    inverse_fn=tf.math.log,
    inverse_log_det_jacobian_fn=(
      lambda y: -tf.reduce_sum(tf.math.log(y), axis=-1)),
  name="LogNormalTransformedDistribution")

Another example constructing a Normal from a StandardNormal:

ds = tfp.distributions
normal = ds.TransformedDistribution(
  distribution=ds.Normal(loc=0., scale=1.),
  bijector=ds.bijectors.Affine(
    shift=-1.,
    scale_identity_multiplier=2.)
  name="NormalTransformedDistribution")

A TransformedDistribution's batch- and event-shape are implied by the base distribution unless explicitly overridden by batch_shape or event_shape arguments. Specifying an overriding batch_shape (event_shape) is permitted only if the base distribution has scalar batch-shape (event-shape). The bijector is applied to the distribution as if the distribution possessed the overridden shape(s). The following example demonstrates how to construct a multivariate Normal as a TransformedDistribution.

ds = tfp.distributions
# We will create two MVNs with batch_shape = event_shape = 2.
mean = [[-1., 0],      # batch:0
        [0., 1]]       # batch:1
chol_cov = [[[1., 0],
             [0, 1]],  # batch:0
            [[1, 0],
             [2, 2]]]  # batch:1
mvn1 = ds.TransformedDistribution(
    distribution=ds.Normal(loc=0., scale=1.),
    bijector=ds.bijectors.Affine(shift=mean, scale_tril=chol_cov),
    batch_shape=[2],  # Valid because base_distribution.batch_shape == [].
    event_shape=[2])  # Valid because base_distribution.event_shape == [].
mvn2 = ds.MultivariateNormalTriL(loc=mean, scale_tril=chol_cov)
# mvn1.log_prob(x) == mvn2.log_prob(x)

Args
`distribution`	The base distribution instance to transform. Typically an instance of `Distribution`.
`bijector`	The object responsible for calculating the transformation. Typically an instance of `Bijector`. `None` means `Identity()`.
`batch_shape`	`integer` vector `Tensor` which overrides `distribution` `batch_shape`; valid only if `distribution.is_scalar_batch()`.
`event_shape`	`integer` vector `Tensor` which overrides `distribution` `event_shape`; valid only if `distribution.is_scalar_event()`.
`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.
`name`	Python `str` name prefixed to Ops created by this class. Default: `bijector.name + distribution.name`.

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.
`bijector`	Function transforming x => y.
`distribution`	Base distribution, p(x).
`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.
`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 `distributions.FULLY_REPARAMETERIZED` or `distributions.NOT_REPARAMETERIZED`.
`validate_args`	Python `bool` indicating possibly expensive checks are enabled.

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'
)

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.

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.

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'
)

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.

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), (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.
`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 (Shanon) cross entropy.

`entropy`

View source

entropy(
    name='entropy'
)

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`.

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

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.

tf.contrib.distributions.TransformedDistribution

Args

Attributes

Methods

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

is_scalar_batch

is_scalar_event

kl_divergence

log_cdf

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`