# tf.contrib.distributions.VectorLaplaceDiag

## Class VectorLaplaceDiag

The vectorization of the Laplace distribution on R^k.

The vector laplace distribution is defined over R^k, and parameterized by a (batch of) length-k loc vector (the means) and a (batch of) k x k scale matrix: covariance = 2 * scale @ scale.T, where @ denotes matrix-multiplication.

#### Mathematical Details

The probability density function (pdf) is,

pdf(x; loc, scale) = exp(-||y||_1) / Z,
y = inv(scale) @ (x - loc),
Z = 2**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||_1 denotes the l1 norm of y, sum_i |y_i|.

A (non-batch) scale matrix is:

scale = diag(scale_diag + scale_identity_multiplier * ones(k))


where:

• scale_diag.shape = [k], and,
• scale_identity_multiplier.shape = [].

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

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

X = (X_1, ..., X_k), each X_i ~ Laplace(loc=0, scale=1)
Y = (Y_1, ...,Y_k) = scale @ X + loc


#### About VectorLaplace and Vector distributions in TensorFlow.

The VectorLaplace is a non-standard distribution that has useful properties.

The marginals Y_1, ..., Y_k are not Laplace random variables, due to the fact that the sum of Laplace random variables is not Laplace.

Instead, Y is a vector whose components are linear combinations of Laplace random variables. Thus, Y lives in the vector space generated by vectors of Laplace distributions. This allows the user to decide the mean and covariance (by setting loc and scale), while preserving some properties of the Laplace distribution. In particular, the tails of Y_i will be (up to polynomial factors) exponentially decaying.

To see this last statement, note that the pdf of Y_i is the convolution of the pdf of k independent Laplace random variables. One can then show by induction that distributions with exponential (up to polynomial factors) tails are closed under convolution.

#### Examples

tfd = tf.contrib.distributions

# Initialize a single 2-variate VectorLaplace.
vla = tfd.VectorLaplaceDiag(
loc=[1., -1],
scale_diag=[1, 2.])

vla.mean().eval()
# ==> [1., -1]

vla.stddev().eval()
# ==> [1., 2] * sqrt(2)

# Evaluate this on an observation in R^2, returning a scalar.
vla.prob([-1., 0]).eval()  # shape: []

# Initialize a 3-batch, 2-variate scaled-identity VectorLaplace.
vla = tfd.VectorLaplaceDiag(
loc=[1., -1],
scale_identity_multiplier=[1, 2., 3])

vla.mean().eval()  # shape: [3, 2]
# ==> [[1., -1]
#      [1, -1],
#      [1, -1]]

vla.stddev().eval()  # shape: [3, 2]
# ==> sqrt(2) * [[1., 1],
#                [2, 2],
#                [3, 3]]

# Evaluate this on an observation in R^2, returning a length-3 vector.
vla.prob([-1., 0]).eval()  # shape: [3]

# Initialize a 2-batch of 3-variate VectorLaplace's.
vla = tfd.VectorLaplaceDiag(
loc=[[1., 2, 3],
[11, 22, 33]]           # shape: [2, 3]
scale_diag=[[1., 2, 3],
[0.5, 1, 1.5]])  # shape: [2, 3]

# Evaluate this on a two observations, each in R^3, returning a length-2
# vector.
x = [[-1., 0, 1],
[-11, 0, 11.]]   # shape: [2, 3].
vla.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.

### 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_diag=None,
scale_identity_multiplier=None,
validate_args=False,
allow_nan_stats=True,
name='VectorLaplaceDiag'
)


Construct Vector Laplace 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 = 2 * scale @ scale.T.

scale = diag(scale_diag + scale_identity_multiplier * ones(k))


where:

• scale_diag.shape = [k], and,
• scale_identity_multiplier.shape = [].

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

#### 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.
• 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.

### 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.

#### 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.

#### 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.

#### 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 VectorLaplaceLinearOperator:

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

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 VectorLaplaceLinearOperator:

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

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()`.