tfp.distributions.TransformedDistribution

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A Transformed Distribution.

Inherits From: Distribution

tfp.distributions.TransformedDistribution(
    distribution, bijector, batch_shape=None, event_shape=None,
    kwargs_split_fn=_default_kwargs_split_fn, validate_args=False, parameters=None,
    name=None
)

Used in the notebooks

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:

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- 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.

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.Normal(loc=0., scale=1.),
    bijector=tfb.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)

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
True
list(b.submodules) == [c]
True
list(c.submodules) == []
True

Methods

batch_shape_tensor

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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.

cdf

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

Cumulative distribution function.

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

cdf(x) := P[X <= x]

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

covariance

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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.

cross_entropy

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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.

entropy

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

Shannon entropy in nats.

event_shape_tensor

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event_shape_tensor(
    name='event_shape_tensor'
)

Shape of a single sample from a single batch as a 1-D int32 Tensor.

is_scalar_batch

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is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

is_scalar_event

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is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

kl_divergence

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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.

log_cdf

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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.

log_prob

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

Log probability density/mass function.

log_survival_function

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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.

mean

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

Mean.

mode

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

Mode.

param_shapes

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@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.

param_static_shapes

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@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.

prob

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

Probability density/mass function.

quantile

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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

sample

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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.

stddev

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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.

survival_function

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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).

variance

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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.

with_name_scope

@classmethod
with_name_scope(
    method
)

Decorator to automatically enter the module name scope.

class MyModule(tf.Module):
  @tf.Module.with_name_scope
  def __call__(self, x):
    if not hasattr(self, 'w'):
      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
    return tf.matmul(x, self.w)

Using the above module would produce tf.Variables and tf.Tensors whose names included the module name:

mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

__getitem__

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__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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.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]

__iter__

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__iter__()