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tf.contrib.distributions.PoissonLogNormalQuadratureCompound

Class `PoissonLogNormalQuadratureCompound`

Inherits From: Distribution

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

PoissonLogNormalQuadratureCompound distribution.

The PoissonLogNormalQuadratureCompound is an approximation to a Poisson-LogNormal compound distribution, i.e.,

p(k|loc, scale)
= int_{R_+} dl LogNormal(l | loc, scale) Poisson(k | l)
approx= sum{ prob[d] Poisson(k | lambda(grid[d])) : d=0, ..., deg-1 }

By default, the grid is chosen as quantiles of the LogNormal distribution parameterized by loc, scale and the prob vector is [1. / quadrature_size]*quadrature_size.

In the non-approximation case, a draw from the LogNormal prior represents the Poisson rate parameter. Unfortunately, the non-approximate distribution lacks an analytical probability density function (pdf). Therefore the PoissonLogNormalQuadratureCompound class implements an approximation based on quadrature.

Note: although the PoissonLogNormalQuadratureCompound is approximately the Poisson-LogNormal compound distribution, it is itself a valid distribution. Viz., it possesses a sample, log_prob, mean, variance, etc. which are all mutually consistent.

Mathematical Details

The PoissonLogNormalQuadratureCompound approximates a Poisson-LogNormal compound distribution. Using variable-substitution and numerical quadrature (default: based on LogNormal quantiles) we can redefine the distribution to be a parameter-less convex combination of deg different Poisson samples.

That is, defined over positive integers, this distribution is parameterized by a (batch of) loc and scale scalars.

The probability density function (pdf) is,

pdf(k | loc, scale, deg)
  = sum{ prob[d] Poisson(k | lambda=exp(grid[d]))
        : d=0, ..., deg-1 }

Examples

tfd = tf.contrib.distributions

# Create two batches of PoissonLogNormalQuadratureCompounds, one with
# prior `loc = 0.` and another with `loc = 1.` In both cases `scale = 1.`
pln = tfd.PoissonLogNormalQuadratureCompound(
    loc=[0., -0.5],
    scale=1.,
    quadrature_size=10,
    validate_args=True)

## Properties

<h3 id="allow_nan_stats"><code>allow_nan_stats</code></h3>

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:

* <b>`allow_nan_stats`</b>: Python `bool`.

<h3 id="batch_shape"><code>batch_shape</code></h3>

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:

* <b>`batch_shape`</b>: `TensorShape`, possibly unknown.

<h3 id="distribution"><code>distribution</code></h3>

Base Poisson parameterized by a quadrature grid.

<h3 id="dtype"><code>dtype</code></h3>

The `DType` of `Tensor`s handled by this `Distribution`.

<h3 id="event_shape"><code>event_shape</code></h3>

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

May be partially defined or unknown.

#### Returns:

* <b>`event_shape`</b>: `TensorShape`, possibly unknown.

<h3 id="loc"><code>loc</code></h3>

Location parameter of the LogNormal prior.

<h3 id="mixture_distribution"><code>mixture_distribution</code></h3>

Distribution which randomly selects a Poisson with quadrature param.

<h3 id="name"><code>name</code></h3>

Name prepended to all ops created by this `Distribution`.

<h3 id="parameters"><code>parameters</code></h3>

Dictionary of parameters used to instantiate this `Distribution`.

<h3 id="quadrature_size"><code>quadrature_size</code></h3>

<h3 id="reparameterization_type"><code>reparameterization_type</code></h3>

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`.

<h3 id="scale"><code>scale</code></h3>

Scale parameter of the LogNormal prior.

<h3 id="validate_args"><code>validate_args</code></h3>

Python `bool` indicating possibly expensive checks are enabled.

## Methods

<h3 id="__init__"><code>__init__</code></h3>

``` python
__init__(
    loc,
    scale,
    quadrature_size=8,
    quadrature_fn=tf.contrib.distributions.quadrature_scheme_lognormal_quantiles,
    validate_args=False,
    allow_nan_stats=True,
    name='PoissonLogNormalQuadratureCompound'
)

Constructs the PoissonLogNormalQuadratureCompound`.

Note: probs returned by (optional) quadrature_fn are presumed to be either a length-quadrature_size vector or a batch of vectors in 1-to-1 correspondence with the returned grid. (I.e., broadcasting is only partially supported.)

Args:

loc: float-like (batch of) scalar Tensor; the location parameter of the LogNormal prior.
scale: float-like (batch of) scalar Tensor; the scale parameter of the LogNormal prior.
quadrature_size: Python int scalar representing the number of quadrature points.
quadrature_fn: Python callable taking loc, scale, quadrature_size, validate_args and returning tuple(grid, probs) representing the LogNormal grid and corresponding normalized weight. normalized) weight. Default value: quadrature_scheme_lognormal_quantiles.
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:

TypeError: if quadrature_grid and quadrature_probs have different base dtype.

`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.

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.

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/PoissonLogNormalQuadratureCompound

tf.contrib.distributions.PoissonLogNormalQuadratureCompound

Class PoissonLogNormalQuadratureCompound

Mathematical Details

Examples

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:

stddev

Args:

Returns:

survival_function

Args:

Returns:

variance

Args:

Returns:

Class `PoissonLogNormalQuadratureCompound`

`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`