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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 theBijector
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
|
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
|
name to give to the op |
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 .
|
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
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
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
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
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
|
name to give to the op |
Returns | |
---|---|
event_shape
|
Tensor .
|
is_scalar_batch
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
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
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
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.
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