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`PoissonLogNormalQuadratureCompound` distribution.

Inherits From: `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

``````import tensorflow_probability as tfp
tfd = tfp.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)

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<tr>
<td>
`loc`
</td>
<td>
`float`-like (batch of) scalar `Tensor`; the location parameter of
the LogNormal prior.
</td>
</tr><tr>
<td>
`scale`
</td>
<td>
`float`-like (batch of) scalar `Tensor`; the scale parameter of
the LogNormal prior.
</td>
</tr><tr>
<td>
</td>
<td>
Python `int` scalar representing the number of quadrature
points.
</td>
</tr><tr>
<td>
</td>
<td>
Python callable taking `loc`, `scale`,
`quadrature_size`, `validate_args` and returning `tuple(grid, probs)`
representing the LogNormal grid and corresponding normalized weight.
normalized) weight.
</td>
</tr><tr>
<td>
`validate_args`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`allow_nan_stats`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
Python `str` name prefixed to Ops created by this class.
</td>
</tr>
</table>

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<tr>
<td>
`TypeError`
</td>
<td>
`dtype`.
</td>
</tr>
</table>

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<tr>
<td>
`allow_nan_stats`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`batch_shape`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`distribution`
</td>
<td>
Base Poisson parameterized by a quadrature grid.
</td>
</tr><tr>
<td>
`dtype`
</td>
<td>
The `DType` of `Tensor`s handled by this `Distribution`.
</td>
</tr><tr>
<td>
`event_shape`
</td>
<td>
Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.
</td>
</tr><tr>
<td>
`loc`
</td>
<td>
Location parameter of the LogNormal prior.
</td>
</tr><tr>
<td>
`mixture_distribution`
</td>
<td>
Distribution which randomly selects a Poisson with quadrature param.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
Name prepended to all ops created by this `Distribution`.
</td>
</tr><tr>
<td>
`parameters`
</td>
<td>
Dictionary of parameters used to instantiate this `Distribution`.
</td>
</tr><tr>
<td>
</td>
<td>

</td>
</tr><tr>
<td>
`reparameterization_type`
</td>
<td>
Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances
`distributions.FULLY_REPARAMETERIZED`
or `distributions.NOT_REPARAMETERIZED`.
</td>
</tr><tr>
<td>
`scale`
</td>
<td>
Scale parameter of the LogNormal prior.
</td>
</tr><tr>
<td>
`validate_args`
</td>
<td>
Python `bool` indicating possibly expensive checks are enabled.
</td>
</tr>
</table>

## Methods

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

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L637-L654">View source</a>

<code>batch_shape_tensor(
name='batch_shape_tensor'
)
</code></pre>

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.

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
name to give to the op
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>

<tr>
<td>
`batch_shape`
</td>
<td>
`Tensor`.
</td>
</tr>
</table>

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

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L881-L898">View source</a>

<code>cdf(
value, name='cdf'
)
</code></pre>

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

```none
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`

View source

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`

View source

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`

View source

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`

View source

Shannon entropy in nats.

### `event_shape_tensor`

View source

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`

View source

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`

View source

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`

View source

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`

View source

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`

View source

Log probability density/mass function.

Args
`value` `float` or `double` `Tensor`.
`name` Python `str` prepended to names of ops created by this function.