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tf.contrib.distributions.VectorExponentialDiag

Class VectorExponentialDiag

The vectorization of the Exponential distribution on R^k.

The vector exponential distribution is defined over a subset of R^k, and parameterized by a (batch of) length-k loc vector and a (batch of) k x k scale matrix: covariance = scale @ scale.T, where @ denotes matrix-multiplication.

Mathematical Details

The probability density function (pdf) is defined over the image of the scale matrix + loc, applied to the positive half-space: Supp = {loc + scale @ x : x in R^k, x_1 > 0, ..., x_k > 0}. On this set,

pdf(y; loc, scale) = exp(-||x||_1) / Z,  for y in Supp
x = inv(scale) @ (y - loc),
Z = |det(scale)|,

where:

• loc is a vector in R^k,
• scale is a linear operator in R^{k x k}, cov = scale @ scale.T,
• Z denotes the normalization constant, and,
• ||x||_1 denotes the l1 norm of x, sum_i |x_i|.

The VectorExponential distribution is a member of the location-scale family, i.e., it can be constructed as,

X = (X_1, ..., X_k), each X_i ~ Exponential(rate=1)
Y = (Y_1, ...,Y_k) = scale @ X + loc

About VectorExponential and Vector distributions in TensorFlow.

The VectorExponential is a non-standard distribution that has useful properties.

The marginals Y_1, ..., Y_k are not Exponential random variables, due to the fact that the sum of Exponential random variables is not Exponential.

Instead, Y is a vector whose components are linear combinations of Exponential random variables. Thus, Y lives in the vector space generated by vectors of Exponential distributions. This allows the user to decide the mean and covariance (by setting loc and scale), while preserving some properties of the Exponential distribution. In particular, the tails of Y_i will be (up to polynomial factors) exponentially decaying.

To see this last statement, note that the pdf of Y_i is the convolution of the pdf of k independent Exponential random variables. One can then show by induction that distributions with exponential (up to polynomial factors) tails are closed under convolution.

Examples

import tensorflow_probability as tfp
tfd = tfp.distributions

# Initialize a single 2-variate VectorExponential, supported on
# {(x, y) in R^2 : x > 0, y > 0}.

# The first component has pdf exp{-x}, the second 0.5 exp{-x / 2}
vex = tfd.VectorExponentialDiag(scale_diag=[1., 2.])

# Compute the pdf of an`R^2` observation; return a scalar.
vex.prob([3., 4.]).eval()  # shape: []

# Initialize a 2-batch of 3-variate Vector Exponential's.
loc = [[1., 2, 3],
[1., 0, 0]]              # shape: [2, 3]
scale_diag = [[1., 2, 3],
[0.5, 1, 1.5]]     # shape: [2, 3]

vex = tfd.VectorExponentialDiag(loc, scale_diag)

# Compute the pdf of two `R^3` observations; return a length-2 vector.
x = [[1.9, 2.2, 3.1],
[10., 1.0, 9.0]]     # shape: [2, 3]
vex.prob(x).eval()    # shape: [2]

__init__

View source

__init__(
loc=None,
scale_diag=None,
scale_identity_multiplier=None,
validate_args=False,
allow_nan_stats=True,
name='VectorExponentialDiag'
)

Construct Vector Exponential distribution supported on a subset of R^k. (deprecated)

The batch_shape is the broadcast shape between loc and scale arguments.

The event_shape is given by last dimension of the matrix implied by scale. The last dimension of loc (if provided) must broadcast with this.

Recall that covariance = scale @ scale.T.

scale = diag(scale_diag + scale_identity_multiplier * ones(k))

where:

• scale_diag.shape = [k], and,
• scale_identity_multiplier.shape = [].

If both scale_diag and scale_identity_multiplier are None, then scale is the Identity matrix.

Args:

• loc: Floating-point Tensor. If this is set to None, loc is implicitly 0. When specified, may have shape [B1, ..., Bb, k] where b >= 0 and k is the event size.
• scale_diag: Non-zero, floating-point Tensor representing a diagonal matrix added to scale. May have shape [B1, ..., Bb, k], b >= 0, and characterizes b-batches of k x k diagonal matrices added to scale. When both scale_identity_multiplier and scale_diag are None then scale is the Identity.
• scale_identity_multiplier: Non-zero, floating-point Tensor representing a scaled-identity-matrix added to scale. May have shape [B1, ..., Bb], b >= 0, and characterizes b-batches of scaled k x k identity matrices added to scale. When both scale_identity_multiplier and scale_diag are None then scale is the Identity.
• 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.
• allow_nan_stats: Python bool, default True. When True, statistics (e.g., mean, mode, variance) use the value "NaN" to indicate the result is undefined. When False, an exception is raised if one or more of the statistic's batch members are undefined.
• name: Python str name prefixed to Ops created by this class.

Raises:

• ValueError: if at most scale_identity_multiplier is specified.

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: Python bool.

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 Tensors 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.

loc

The loc Tensor in Y = scale @ X + loc.

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.

scale

The scale LinearOperator in Y = scale @ X + loc.

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

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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

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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), (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

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entropy(name='entropy')

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.

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

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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 (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

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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.

Returns:

• logcdf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_prob

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log_prob(
value,
name='log_prob'
)

Log probability density/mass function.

value is a batch vector with compatible shape if value is a Tensor whose shape can be broadcast up to either:

self.batch_shape + self.event_shape

or

[M1, ..., Mm] + self.batch_shape + self.event_shape

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

Returns:

• log_prob: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_survival_function

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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 or double Tensor.
• name: Python str prepended to names of ops created by this function.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

mean

View source

mean(name='mean')

Mean.

mode

View source

mode(name='mode')

Mode.

param_shapes

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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 to sample().
• name: name to prepend ops with.

Returns:

dict of parameter name to Tensor shapes.

param_static_shapes

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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 to sample().

Returns:

dict of parameter name to TensorShape.

Raises:

• ValueError: if sample_shape is a TensorShape and is not fully defined.

prob

View source

prob(
value,
name='prob'
)

Probability density/mass function.

value is a batch vector with compatible shape if value is a Tensor whose shape can be broadcast up to either:

self.batch_shape + self.event_shape

or

[M1, ..., Mm] + self.batch_shape + self.event_shape

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

Returns:

• prob: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

quantile

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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 or double Tensor.
• name: Python str prepended to names of ops created by this function.

Returns:

• quantile: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

sample

View source

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 1D int32 Tensor. Shape of the generated samples.
• seed: Python integer seed for RNG
• name: name to give to the op.

Returns:

• samples: a Tensor with prepended dimensions sample_shape.

stddev

View source

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: Python str prepended to names of ops created by this function.

Returns:

• stddev: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

survival_function

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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 or double Tensor.
• name: Python str prepended to names of ops created by this function.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

variance

View source

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: Python str prepended to names of ops created by this function.

Returns:

• variance: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().