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tf.contrib.distributions.WishartCholesky

Class WishartCholesky

Defined in tensorflow/contrib/distributions/python/ops/wishart.py.

See the guide: Statistical Distributions (contrib) > Multivariate distributions

The matrix Wishart distribution on positive definite matrices.

This distribution is defined by a scalar degrees of freedom df and a lower, triangular Cholesky factor which characterizes the scale matrix.

Using WishartCholesky is a constant-time improvement over WishartFull. It saves an O(nbk^3) operation, i.e., a matrix-product operation for sampling and a Cholesky factorization in log_prob. For most use-cases it often saves another O(nbk^3) operation since most uses of Wishart will also use the Cholesky factorization.

Mathematical Details

The probability density function (pdf) is,

pdf(X; df, scale) = det(X)**(0.5 (df-k-1)) exp(-0.5 tr[inv(scale) X]) / Z
Z = 2**(0.5 df k) |det(scale)|**(0.5 df) Gamma_k(0.5 df)

where: df >= k denotes the degrees of freedom, scale is a symmetric, positive definite, k x k matrix, Z is the normalizing constant, and, Gamma_k is the multivariate Gamma function.

Examples

# Initialize a single 3x3 Wishart with Cholesky factored scale matrix and 5
# degrees-of-freedom.(*)
df = 5
chol_scale = tf.cholesky(...)  # Shape is [3, 3].
dist = tf.contrib.distributions.WishartCholesky(df=df, scale=chol_scale)

# Evaluate this on an observation in R^3, returning a scalar.
x = ...  # A 3x3 positive definite matrix.
dist.prob(x)  # Shape is [], a scalar.

# Evaluate this on a two observations, each in R^{3x3}, returning a length two
# Tensor.
x = [x0, x1]  # Shape is [2, 3, 3].
dist.prob(x)  # Shape is [2].

# Initialize two 3x3 Wisharts with Cholesky factored scale matrices.
df = [5, 4]
chol_scale = tf.cholesky(...)  # Shape is [2, 3, 3].
dist = tf.contrib.distributions.WishartCholesky(df=df, scale=chol_scale)

# Evaluate this on four observations.
x = [[x0, x1], [x2, x3]]  # Shape is [2, 2, 3, 3].
dist.prob(x)  # Shape is [2, 2].

# (*) - To efficiently create a trainable covariance matrix, see the example
#   in tf.contrib.distributions.matrix_diag_transform.

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.

cholesky_input_output_matrices

Boolean indicating if Tensor input/outputs are Cholesky factorized.

df

Wishart distribution degree(s) of freedom.

dimension

Dimension of underlying vector space. The p in R^(p*p).

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.

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_operator

Wishart distribution scale matrix as an Linear Operator.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

__init__

__init__(
    df,
    scale,
    cholesky_input_output_matrices=False,
    validate_args=False,
    allow_nan_stats=True,
    name='WishartCholesky'
)

Construct Wishart distributions.

Args:

  • df: float or double Tensor. Degrees of freedom, must be greater than or equal to dimension of the scale matrix.
  • scale: float or double Tensor. The Cholesky factorization of the symmetric positive definite scale matrix of the distribution.
  • cholesky_input_output_matrices: Python bool. Any function which whose input or output is a matrix assumes the input is Cholesky and returns a Cholesky factored matrix. Example log_prob input takes a Cholesky and sample_n returns a Cholesky when cholesky_input_output_matrices=True.
  • 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.

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.

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 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:

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:

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.

Returns:

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

log_normalization

log_normalization(name='log_normalization')

Computes the log normalizing constant, log(Z).

log_prob

log_prob(
    value,
    name='log_prob'
)

Log probability density/mass function.

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

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

mean(name='mean')

Mean.

mean_log_det

mean_log_det(name='mean_log_det')

Computes E[log(det(X))] under this Wishart distribution.

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 to sample().
  • 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 to sample().

Returns:

dict of parameter name to TensorShape.

Raises:

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

prob

prob(
    value,
    name='prob'
)

Probability density/mass function.

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

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

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.

scale

scale()

Wishart distribution scale matrix.

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: 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

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

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().

© 2018 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/WishartCholesky