tf.contrib.distributions.WishartCholesky

class tf.contrib.distributions.WishartCholesky

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 PDF of this distribution is,

f(X) = det(X)^(0.5 (df-k-1)) exp(-0.5 tr[inv(scale) X]) / B(scale, df)

where df >= k denotes the degrees of freedom, scale is a symmetric, pd, k x k matrix, and the normalizing constant B(scale, df) is given by:

B(scale, df) = 2^(0.5 df k) |det(scale)|^(0.5 df) Gamma_k(0.5 df)

where 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.pdf(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.pdf(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.pdf(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 boolean 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 boolean.

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.

is_continuous

is_reparameterized

name

Name prepended to all ops created by this Distribution.

parameters

Dictionary of parameters used to instantiate this Distribution.

scale_operator_pd

Wishart distribution scale matrix as an OperatorPD.

validate_args

Python boolean indicated possibly expensive checks are enabled.

Methods

__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: Boolean. Any function which whose input or output is a matrix assumes the input is Cholesky and returns a Cholesky factored matrix. Examplelog_pdf input takes a Cholesky and sample_n returns a Cholesky when cholesky_input_output_matrices=True.
  • validate_args: Boolean, default False. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
  • allow_nan_stats: Boolean, default True. If False, raise an exception if a statistic (e.g., mean, mode) is undefined for any batch member. If True, batch members with valid parameters leading to undefined statistics will return NaN for this statistic.
  • name: The name scope to give class member ops.

batch_shape(name='batch_shape')

Shape of a single sample from a single event index as a 1-D Tensor.

The product of the dimensions of the batch_shape is the number of independent distributions of this kind the instance represents.

Args:

  • name: name to give to the op

Returns:

  • batch_shape: Tensor.

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

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: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

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) intitialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

entropy(name='entropy')

Shannon entropy in nats.

event_shape(name='event_shape')

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.

get_batch_shape()

Shape of a single sample from a single event index as a TensorShape.

Same meaning as batch_shape. May be only partially defined.

Returns:

  • batch_shape: TensorShape, possibly unknown.

get_event_shape()

Shape of a single sample from a single batch as a TensorShape.

Same meaning as event_shape. May be only partially defined.

Returns:

  • event_shape: TensorShape, possibly unknown.

is_scalar_batch(name='is_scalar_batch')

Indicates that batch_shape == [].

Args:

  • name: The name to give this op.

Returns:

  • is_scalar_batch: Boolean scalar Tensor.

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

  • name: The name to give this op.

Returns:

  • is_scalar_event: Boolean scalar Tensor.

log_cdf(value, name='log_cdf', **condition_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.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

log_normalizing_constant(name='log_normalizing_constant')

Computes the log normalizing constant, log(Z).

log_pdf(value, name='log_pdf', **condition_kwargs)

Log probability density function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

  • TypeError: if not is_continuous.

log_pmf(value, name='log_pmf', **condition_kwargs)

Log probability mass function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

  • TypeError: if is_continuous.

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

Log probability density/mass function (depending on is_continuous).

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

log_survival_function(value, name='log_survival_function', **condition_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.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

mean(name='mean')

Mean.

mean_log_det(name='mean_log_det')

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

mode(name='mode')

Mode.

param_shapes(cls, sample_shape, name='DistributionParamShapes')

Shapes of parameters given the desired shape of a call to sample().

Subclasses should override static 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(cls, sample_shape)

param_shapes with static (i.e. TensorShape) shapes.

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.

pdf(value, name='pdf', **condition_kwargs)

Probability density function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

  • TypeError: if not is_continuous.

pmf(value, name='pmf', **condition_kwargs)

Probability mass function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

  • TypeError: if is_continuous.

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

Probability density/mass function (depending on is_continuous).

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

sample(sample_shape=(), seed=None, name='sample', **condition_kwargs)

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. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • samples: a Tensor with prepended dimensions sample_shape.

scale()

Wishart distribution scale matrix.

std(name='std')

Standard deviation.

survival_function(value, name='survival_function', **condition_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).

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

Tensorof shapesample_shape(x) + self.batch_shapewith values of typeself.dtype`.

variance(name='variance')

Variance.

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