tfp.substrates.jax.distributions.TransformedDistribution

A Transformed Distribution.

Inherits From: Distribution

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.

Kullback-Leibler divergence is also well defined for TransformedDistribution instances that have matching bijectors. Bijector matching is performed via the Bijector.eq method, e.g., td1.bijector == td2.bijector, as part of the KL calculation. If the underlying bijectors do not match, a NotImplementedError is raised when calling kl_divergence. This is the same behavior as calling kl_divergence when two distributions do not have a registered KL divergence.

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

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

A LogNormal made from callables:

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

Another example constructing a Normal from a StandardNormal:

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

A TransformedDistribution's batch_shape is the same as that of the base distribution, and its event_shape is the forward_event_shape of the bijector applied to the event_shape of the base distribution.

tfd.Sample or tfd.Independent may be used to add extra IID dimensions to the event_shape of the base distribution before the bijector operates on it. The following example demonstrates how to construct a multivariate Normal as a TransformedDistribution, by adding a rank-1 IID dimension to the event_shape of a standard Normal and applying tfb.ScaleMatvecTriL.

tfd = tfp.distributions
tfb = tfp.bijectors
# 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 = tfd.TransformedDistribution(
    distribution=tfd.Sample(
        tfd.Normal(loc=[0., 0], scale=1.),  # base_dist.batch_shape == [2]
        sample_shape=[2])                   # base_dist.event_shape == [2]
    bijector=tfb.Shift(shift=mean)(tfb.ScaleMatvecTriL(scale_tril=chol_cov)))
mvn2 = ds.MultivariateNormalTriL(loc=mean, scale_tril=chol_cov)
# mvn1.log_prob(x) == mvn2.log_prob(x)

<!-- Tabular view -->
 <table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2"><h2 class="add-link">Args</h2></th></tr>

<tr>
<td>
`distribution`
</td>
<td>
The base distribution instance to transform. Typically an
instance of `Distribution`.
</td>
</tr><tr>
<td>
`bijector`
</td>
<td>
The object responsible for calculating the transformation.
Typically an instance of `Bijector`.
</td>
</tr><tr>
<td>
`kwargs_split_fn`
</td>
<td>
Python `callable` which takes a kwargs `dict` and returns
a tuple of kwargs `dict`s for each of the `distribution` and `bijector`
parameters respectively.
Default value: `_default_kwargs_split_fn` (i.e.,
`lambda kwargs: (kwargs.get('distribution_kwargs', {}),
kwargs.get('bijector_kwargs', {}))`)
</td>
</tr><tr>
<td>
`validate_args`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`parameters`
</td>
<td>
Locals dict captured by subclass constructor, to be used for
copy/slice re-instantiation operations.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
Python `str` name prefixed to Ops created by this class. Default:
`bijector.name + distribution.name`.
</td>
</tr>
</table>





<!-- Tabular view -->
 <table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2"><h2 class="add-link">Attributes</h2></th></tr>

<tr>
<td>
`allow_nan_stats`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`batch_shape`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`bijector`
</td>
<td>
Function transforming x => y.
</td>
</tr><tr>
<td>
`distribution`
</td>
<td>
Base distribution, p(x).
</td>
</tr><tr>
<td>
`dtype`
</td>
<td>
The `DType` of `Tensor`s handled by this `Distribution`.
</td>
</tr><tr>
<td>
`event_shape`
</td>
<td>
Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
Name prepended to all ops created by this `Distribution`.
</td>
</tr><tr>
<td>
`parameters`
</td>
<td>
Dictionary of parameters used to instantiate this `Distribution`.
</td>
</tr><tr>
<td>
`reparameterization_type`
</td>
<td>
Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances
`tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.
</td>
</tr><tr>
<td>
`trainable_variables`
</td>
<td>

</td>
</tr><tr>
<td>
`validate_args`
</td>
<td>
Python `bool` indicating possibly expensive checks are enabled.
</td>
</tr><tr>
<td>
`variables`
</td>
<td>

</td>
</tr>
</table>



## Methods

<h3 id="batch_shape_tensor"><code>batch_shape_tensor</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/master/tensorflow_probability/substrates/jax/distributions/distribution.py">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>batch_shape_tensor(
    name='batch_shape_tensor'
)
</code></pre>

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.

<!-- Tabular view -->
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<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
name to give to the op
</td>
</tr>
</table>



<!-- Tabular view -->
 <table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Returns</th></tr>

<tr>
<td>
`batch_shape`
</td>
<td>
`Tensor`.
</td>
</tr>
</table>



<h3 id="cdf"><code>cdf</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/master/tensorflow_probability/substrates/jax/distributions/distribution.py">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>cdf(
    value, name='cdf', **kwargs
)
</code></pre>

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

```none
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

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.

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

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: Chi, ExpInverseGamma, GeneralizedExtremeValue, Gumbel, JohnsonSU, Kumaraswamy, LogLogistic, LogNormal, LogitNormal, Moyal, MultivariateNormalDiag, MultivariateNormalDiagPlusLowRank, MultivariateNormalFullCovariance, MultivariateNormalLinearOperator, MultivariateNormalTriL, RelaxedOneHotCategorical, SinhArcsinh, TransformedDistribution, VectorExponentialDiag, Weibull

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

Shannon entropy in nats.

event_shape_tensor

View source

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

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.

is_scalar_batch

View source

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

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

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: Chi, ExpInverseGamma, GeneralizedExtremeValue, Gumbel, JohnsonSU, Kumaraswamy, LogLogistic, LogNormal, LogitNormal, Moyal, MultivariateNormalDiag, MultivariateNormalDiagPlusLowRank, MultivariateNormalFullCovariance, MultivariateNormalLinearOperator, MultivariateNormalTriL, RelaxedOneHotCategorical, SinhArcsinh, TransformedDistribution, VectorExponentialDiag, Weibull

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 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 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.

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.

mode

View source

Mode.

param_shapes

View source

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

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

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.

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.

prob

View source

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 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

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

View source

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.

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

View source

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

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