## Class `PoissonLogNormalQuadratureCompound`

Inherits From: `Distribution`

`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.

#### 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.`
loc=[0., -0.5],
scale=1.,
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="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,
validate_args=False,
allow_nan_stats=True,
)
``````

THIS FUNCTION IS DEPRECATED. It will be removed after 2018-10-01. Instructions for updating: The TensorFlow Distributions library has moved to TensorFlow Probability (https://github.com/tensorflow/probability). You should update all references to use `tfp.distributions` instead of `tf.contrib.distributions`.

#### 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.

#### 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.

#### 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()`.