contrib.distributions.TransformedDistribution
tf.contrib.distributions.TransformedDistribution
class tf.contrib.distributions.TransformedDistribution
Defined in tensorflow/python/ops/distributions/transformed_distribution.py
.
See the guide: Statistical Distributions (contrib) > Transformed distributions
A Transformed 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
.
A simple example constructing a Log-Normal distribution from a Normal distribution:
ds = tf.contrib.distributions log_normal = ds.TransformedDistribution( distribution=ds.Normal(loc=0., scale=1.), bijector=ds.bijectors.Exp(), name="LogNormalTransformedDistribution")
A LogNormal
made from callables:
ds = tf.contrib.distributions log_normal = ds.TransformedDistribution( distribution=ds.Normal(loc=0., scale=1.), bijector=ds.bijectors.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:
ds = tf.contrib.distributions normal = ds.TransformedDistribution( distribution=ds.Normal(loc=0., scale=1.), bijector=ds.bijectors.Affine( shift=-1., scale_identity_multiplier=2., event_ndims=0), 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 = tf.contrib.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)
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:
-
allow_nan_stats
: Pythonbool
.
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.
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.
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 distributions.FULLY_REPARAMETERIZED
or distributions.NOT_REPARAMETERIZED
.
Returns:
An instance of ReparameterizationType
.
validate_args
Python bool
indicating possibly expensive checks are enabled.
Methods
__init__
__init__( distribution, bijector=None, batch_shape=None, event_shape=None, validate_args=False, name=None )
Construct a Transformed Distribution.
Args:
-
distribution
: The base distribution instance to transform. Typically an instance ofDistribution
. -
bijector
: The object responsible for calculating the transformation. Typically an instance ofBijector
.None
meansIdentity()
. -
batch_shape
:integer
vectorTensor
which overridesdistribution
batch_shape
; valid only ifdistribution.is_scalar_batch()
. -
event_shape
:integer
vectorTensor
which overridesdistribution
event_shape
; valid only ifdistribution.is_scalar_event()
. -
validate_args
: Pythonbool
, defaultFalse
. WhenTrue
distribution parameters are checked for validity despite possibly degrading runtime performance. WhenFalse
invalid inputs may silently render incorrect outputs. -
name
: Pythonstr
name prefixed to Ops created by this class. Default:bijector.name + distribution.name
.
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
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
cdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
copy
copy(**override_parameters_kwargs)
Creates a deep copy of the distribution.
Note: the copy distribution may continue to depend on the original initialization arguments.
Args:
**override_parameters_kwargs: String/value dictionary of initialization arguments to override with new values.
Returns:
-
distribution
: A new instance oftype(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
: The name to give this op.
Returns:
-
covariance
: Floating-pointTensor
with shape[B1, ..., Bn, k', k']
where the firstn
dimensions are batch coordinates andk' = reduce_prod(self.event_shape)
.
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
: The name to give this op.
Returns:
-
is_scalar_batch
:bool
scalarTensor
.
is_scalar_event
is_scalar_event(name='is_scalar_event')
Indicates that event_shape == []
.
Args:
-
name
: The name to give this op.
Returns:
-
is_scalar_event
:bool
scalarTensor
.
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
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
logcdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
log_prob
log_prob( value, name='log_prob' )
Log probability density/mass function.
Args:
-
value
:float
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
log_prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.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
ordouble
Tensor
. -
name
: The name to give this op.
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
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 tosample()
. -
name
: name to prepend ops with.
Returns:
dict
of parameter name to Tensor
shapes.
param_static_shapes
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 tosample()
.
Returns:
dict
of parameter name to TensorShape
.
Raises:
-
ValueError
: ifsample_shape
is aTensorShape
and is not fully defined.
prob
prob( value, name='prob' )
Probability density/mass function.
Args:
-
value
:float
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.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
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
quantile
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.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:
-
sample_shape
: 0D or 1Dint32
Tensor
. Shape of the generated samples. -
seed
: Python integer seed for RNG -
name
: name to give to the op.
Returns:
-
samples
: aTensor
with prepended dimensionssample_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:
-
name
: The name to give this op.
Returns:
-
stddev
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.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
ordouble
Tensor
. -
name
: The name to give this op.
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:
-
name
: The name to give this op.
Returns:
-
variance
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.mean()
.
© 2017 The TensorFlow Authors. All rights reserved.
Licensed under the Creative Commons Attribution License 3.0.
Code samples licensed under the Apache 2.0 License.
https://www.tensorflow.org/api_docs/python/tf/contrib/distributions/TransformedDistribution