tf.contrib.distributions.MultivariateNormalDiagPlusLowRank

The multivariate normal distribution on R^k.

tf.contrib.distributions.MultivariateNormalDiagPlusLowRank(
    loc=None, scale_diag=None, scale_identity_multiplier=None,
    scale_perturb_factor=None, scale_perturb_diag=None, validate_args=False,
    allow_nan_stats=True, name='MultivariateNormalDiagPlusLowRank'
)

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 linear operator in R^{k x k}, cov = 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 = diag(scale_diag + scale_identity_multiplier ones(k)) +
      scale_perturb_factor @ diag(scale_perturb_diag) @ scale_perturb_factor.T

where:

• scale_diag.shape = [k],
• scale_identity_multiplier.shape = [],
• scale_perturb_factor.shape = [k, r], typically k >> r, and,
• scale_perturb_diag.shape = [r].

Additional leading dimensions (if any) will index batches.

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

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

Examples

import tensorflow_probability as tfp
tfd = tfp.distributions

# Initialize a single 3-variate Gaussian with covariance `cov = S @ S.T`,
# `S = diag(d) + U @ diag(m) @ U.T`. The perturbation, `U @ diag(m) @ U.T`, is
# a rank-2 update.
mu = [-0.5., 0, 0.5]   # shape: [3]
d = [1.5, 0.5, 2]      # shape: [3]
U = [[1., 2],
     [-1, 1],
     [2, -0.5]]        # shape: [3, 2]
m = [4., 5]            # shape: [2]
mvn = tfd.MultivariateNormalDiagPlusLowRank(
    loc=mu
    scale_diag=d
    scale_perturb_factor=U,
    scale_perturb_diag=m)

# Evaluate this on an observation in `R^3`, returning a scalar.
mvn.prob([-1, 0, 1]).eval()  # shape: []

# Initialize a 2-batch of 3-variate Gaussians; `S = diag(d) + U @ U.T`.
mu = [[1.,  2,  3],
      [11, 22, 33]]      # shape: [b, k] = [2, 3]
U = [[[1., 2],
      [3,  4],
      [5,  6]],
     [[0.5, 0.75],
      [1,0, 0.25],
      [1.5, 1.25]]]      # shape: [b, k, r] = [2, 3, 2]
m = [[0.1, 0.2],
     [0.4, 0.5]]         # shape: [b, r] = [2, 2]

mvn = tfd.MultivariateNormalDiagPlusLowRank(
    loc=mu,
    scale_perturb_factor=U,
    scale_perturb_diag=m)

mvn.covariance().eval()   # shape: [2, 3, 3]
# ==> [[[  15.63   31.57    48.51]
#       [  31.57   69.31   105.05]
#       [  48.51  105.05   162.59]]
#
#      [[   2.59    1.41    3.35]
#       [   1.41    2.71    3.34]
#       [   3.35    3.34    8.35]]]

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

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.
scale_perturb_factor	Floating-point Tensor representing a rank-r perturbation added to scale. May have shape [B1, ..., Bb, k, r], b >= 0, and characterizes b-batches of rank-r updates to scale. When None, no rank-r update is added to scale.
scale_perturb_diag	Floating-point Tensor representing a diagonal matrix inside the rank-r perturbation added to scale. May have shape [B1, ..., Bb, r], b >= 0, and characterizes b-batches of r x r diagonal matrices inside the perturbation added to scale. When None, an identity matrix is used inside the perturbation. Can only be specified if scale_perturb_factor is also specified.
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.

Attributes
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.
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.
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.
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.
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

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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.

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

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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

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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

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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.

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	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

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mode(
    name='mode'
)

Mode.

param_shapes

View source

@classmethod
param_shapes(
    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

View source

@classmethod
param_static_shapes(
    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

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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	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

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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

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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().