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tf.contrib.distributions.MultivariateNormalTriL

Class `MultivariateNormalTriL`

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

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

The multivariate normal distribution on R^k.

The Multivariate Normal distribution is defined over R^k and parameterized by a (batch of) length-k loc vector (aka "mu") 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,

pdf(x; loc, scale) = exp(-0.5 ||y||**2) / Z,
y = inv(scale) @ (x - loc),
Z = (2 pi)**(0.5 k) |det(scale)|,

where:

loc is a vector in R^k,
scale is a matrix in R^{k x k}, covariance = scale @ scale.T,
Z denotes the normalization constant, and,
||y||**2 denotes the squared Euclidean norm of y.

A (non-batch) scale matrix is:

scale = scale_tril

where scale_tril is lower-triangular k x k matrix with non-zero diagonal, i.e., tf.diag_part(scale_tril) != 0.

Additional leading dimensions (if any) will index batches.

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

X ~ MultivariateNormal(loc=0, scale=1)   # Identity scale, zero shift.
Y = scale @ X + loc

Trainable (batch) lower-triangular matrices can be created with tf.contrib.distributions.matrix_diag_transform() and/or tf.contrib.distributions.fill_triangular()

Examples

tfd = tf.contrib.distributions

# Initialize a single 3-variate Gaussian.
mu = [1., 2, 3]
cov = [[ 0.36,  0.12,  0.06],
       [ 0.12,  0.29, -0.13],
       [ 0.06, -0.13,  0.26]]
scale = tf.cholesky(cov)
# ==> [[ 0.6,  0. ,  0. ],
#      [ 0.2,  0.5,  0. ],
#      [ 0.1, -0.3,  0.4]])
mvn = tfd.MultivariateNormalTriL(
    loc=mu,
    scale_tril=scale)

mvn.mean().eval()
# ==> [1., 2, 3]

# Covariance agrees with cholesky(cov) parameterization.
mvn.covariance().eval()
# ==> [[ 0.36,  0.12,  0.06],
#      [ 0.12,  0.29, -0.13],
#      [ 0.06, -0.13,  0.26]]

# Compute the pdf of an observation in `R^3` ; return a scalar.
mvn.prob([-1., 0, 1]).eval()  # shape: []

# Initialize a 2-batch of 3-variate Gaussians.
mu = [[1., 2, 3],
      [11, 22, 33]]              # shape: [2, 3]
tril = ...  # shape: [2, 3, 3], lower triangular, non-zero diagonal.
mvn = tfd.MultivariateNormalTriL(
    loc=mu,
    scale_tril=tril)

# Compute the pdf of two `R^3` observations; return a length-2 vector.
x = [[-0.9, 0, 0.1],
     [-10, 0, 9]]     # shape: [2, 3]
mvn.prob(x).eval()    # shape: [2]

# Instantiate a "learnable" MVN.
dims = 4
with tf.variable_scope("model"):
  mvn = tfd.MultivariateNormalTriL(
      loc=tf.get_variable(shape=[dims], dtype=tf.float32, name="mu"),
      scale_tril=tfd.fill_triangular(
          tf.get_variable(shape=[dims * (dims + 1) / 2],
                          dtype=tf.float32, name="chol_Sigma")))

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

`init`

__init__(
    loc=None,
    scale_tril=None,
    validate_args=False,
    allow_nan_stats=True,
    name='MultivariateNormalTriL'
)

Construct Multivariate Normal distribution on R^k.

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. A (non-batch) scale matrix is:

scale = scale_tril

where scale_tril is lower-triangular k x k matrix with non-zero diagonal, i.e., tf.diag_part(scale_tril) != 0.

Additional leading dimensions (if any) will index batches.

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_tril: Floating-point, lower-triangular Tensor with non-zero diagonal elements. scale_tril has shape [B1, ..., Bb, k, k] where b >= 0 and k is the event size.
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 neither loc nor scale_tril are specified.

`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:

other: tf.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: tf.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.

Returns:

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

`log_prob`

log_prob(
    value,
    name='log_prob'
)

Log probability density/mass function.

Additional documentation from MultivariateNormalLinearOperator:

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

[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`

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.

`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.

Additional documentation from MultivariateNormalLinearOperator:

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

[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`

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.

`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/MultivariateNormalTriL

tf.contrib.distributions.MultivariateNormalTriL

Class MultivariateNormalTriL

Mathematical Details

Examples

Properties

allow_nan_stats

Returns:

batch_shape

Returns:

bijector

distribution

dtype

event_shape

Returns:

loc

name

parameters

reparameterization_type

Returns:

scale

validate_args

Methods

__init__

Args:

Raises:

batch_shape_tensor

Args:

Returns:

cdf

Args:

Returns:

copy

Args:

Returns:

covariance

Args:

Returns:

cross_entropy

Args:

Returns:

entropy

event_shape_tensor

Args:

Returns:

is_scalar_batch

Args:

Returns:

is_scalar_event

Args:

Returns:

kl_divergence

Args:

Returns:

log_cdf

Args:

Returns:

log_prob

Args:

Returns:

log_survival_function

Args:

Returns:

mean

mode

param_shapes

Args:

Returns:

param_static_shapes

Args:

Returns:

Raises:

prob

Args:

Returns:

quantile

Args:

Returns:

sample

Args:

Returns:

Class `MultivariateNormalTriL`

`allow_nan_stats`

`batch_shape`

`bijector`

`distribution`

`dtype`

`event_shape`

`loc`

`name`

`parameters`

`reparameterization_type`

`scale`

`validate_args`

`init`

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`

`log_prob`

`log_survival_function`

`mean`

`mode`

`param_shapes`

`param_static_shapes`

`prob`

`quantile`

`sample`

`stddev`

`survival_function`

`variance`