# ed.models.MultivariateNormalFullCovariance

## Class MultivariateNormalFullCovariance

Inherits From: RandomVariable

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 covariance_matrix matrices that are the covariance. This is different than the other multivariate normals, which are parameterized by a matrix more akin to the standard deviation.

#### Mathematical Details

The probability density function (pdf) is, with @ as matrix multiplication,

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

where:

• loc is a vector in R^k,
• covariance_matrix is an R^{k x k} symmetric positive definite matrix,
• Z denotes the normalization constant, and,
• ||y||**2 denotes the squared Euclidean norm of y.

Additional leading dimensions (if any) in loc and covariance_matrix allow for batch dimensions.

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

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

#### Examples

ds = 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]]
mvn = ds.MultivariateNormalFullCovariance(
loc=mu,
covariance_matrix=cov)

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

# Covariance agrees with covariance_matrix.
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]
covariance_matrix = ...  # shape: [2, 3, 3], symmetric, positive definite.
mvn = ds.MultivariateNormalFullCovariance(
loc=mu,
covariance=covariance_matrix)

# 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]


## 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.

### sample_shape

Sample shape of random variable.

### scale

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

### shape

Shape of random variable.

### unique_name

Name of random variable with its unique scoping name. Use name to just get the name of the random variable.

### validate_args

Python bool indicating possibly expensive checks are enabled.

## Methods

### init

__init__(
*args,
**kwargs
)

Construct Multivariate Normal distribution on R^k.

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

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

A non-batch covariance_matrix matrix is a k x k symmetric positive definite matrix. In other words it is (real) symmetric with all eigenvalues strictly positive.

#### 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.
• covariance_matrix: Floating-point, symmetric positive definite Tensor of same dtype as loc. The strict upper triangle of covariance_matrix is ignored, so if covariance_matrix is not symmetric no error will be raised (unless validate_args is True). covariance_matrix 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 covariance_matrix are specified.

### abs

__abs__()

### add

__add__(other)

### and

__and__(other)

### bool

__bool__()

### div

__div__(other)

### eq

__eq__(other)

### floordiv

__floordiv__(other)

### ge

__ge__(other)

### getitem

__getitem__(key)

Subset the tensor associated to the random variable, not the random variable itself.

### gt

__gt__(other)

### invert

__invert__()

### iter

__iter__()

### le

__le__(other)

### lt

__lt__(other)

### mod

__mod__(other)

### mul

__mul__(other)

### neg

__neg__()

### nonzero

__nonzero__()

### or

__or__(other)

### pow

__pow__(other)

### radd

__radd__(other)

### rand

__rand__(other)

### rdiv

__rdiv__(other)

### rfloordiv

__rfloordiv__(other)

### rmod

__rmod__(other)

### rmul

__rmul__(other)

### ror

__ror__(other)

### rpow

__rpow__(other)

### rsub

__rsub__(other)

### rtruediv

__rtruediv__(other)

### rxor

__rxor__(other)

### sub

__sub__(other)

### truediv

__truediv__(other)

### xor

__xor__(other)

### 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: The name to give this op.

#### 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: The name to give this op.

#### 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).

### entropy

entropy(name='entropy')

Shannon entropy in nats.

### eval

eval(
session=None,
feed_dict=None
)

In a session, computes and returns the value of this random variable.

This is not a graph construction method, it does not add ops to the graph.

This convenience method requires a session where the graph containing this variable has been launched. If no session is passed, the default session is used.

#### Args:

• session: tf.BaseSession, optional. The tf.Session to use to evaluate this random variable. If none, the default session is used.
• feed_dict: dict, optional. A dictionary that maps tf.Tensor objects to feed values. See tf.Session.run() for a description of the valid feed values.

#### Examples

x = Normal(0.0, 1.0)
with tf.Session() as sess:
# Usage passing the session explicitly.
print(x.eval(sess))
# Usage with the default session.  The 'with' block
# above makes 'sess' the default session.
print(x.eval())

### 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.

### get_ancestors

get_ancestors(collection=None)

Get ancestor random variables.

### get_blanket

get_blanket(collection=None)

Get the random variable's Markov blanket.

### get_children

get_children(collection=None)

Get child random variables.

### get_descendants

get_descendants(collection=None)

Get descendant random variables.

### get_parents

get_parents(collection=None)

Get parent random variables.

### get_shape

get_shape()

Get shape of random variable.

### get_siblings

get_siblings(collection=None)

Get sibling random variables.

### get_variables

get_variables(collection=None)

Get TensorFlow variables that the random variable depends on.

### 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 scalar Tensor.

### 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 scalar Tensor.

### 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: The name to give this op.

#### 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

or

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

#### Args:

• value: float or double Tensor.
• name: The name to give this op.

#### 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: 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 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

or

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

#### Args:

• value: float or double Tensor.
• name: The name to give this op.

#### 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: The name to give this op.

#### 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: The name to give this op.

#### 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: The name to give this op.

#### Returns:

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

### value

value()

Get tensor that the random variable corresponds to.

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