Multinomial distribution.

This distribution is parameterized by a vector `p` of probability
parameters for `k` classes and `n`, the counts per each class..

#### Mathematical details

The Multinomial is a distribution over k-class count data, meaning
for each k-tuple of non-negative integer `counts = [n_1,...,n_k]`, we have a
probability of these draws being made from the distribution.  The distribution
has hyperparameters `p = (p_1,...,p_k)`, and probability mass
function (pmf):

```pmf(counts) = n! / (n_1!...n_k!) * (p_1)^n_1*(p_2)^n_2*...(p_k)^n_k```

where above `n = sum_j n_j`, `n!` is `n` factorial.

#### Examples

Create a 3-class distribution, with the 3rd class is most likely to be drawn,
using logits..

```python
logits = [-50., -43, 0]
dist = Multinomial(n=4., logits=logits)
```

Create a 3-class distribution, with the 3rd class is most likely to be drawn.

```python
p = [.2, .3, .5]
dist = Multinomial(n=4., p=p)
```

The distribution functions can be evaluated on counts.

```python
# counts same shape as p.
counts = [1., 0, 3]
dist.prob(counts)  # Shape []

# p will be broadcast to [[.2, .3, .5], [.2, .3, .5]] to match counts.
counts = [[1., 2, 1], [2, 2, 0]]
dist.prob(counts)  # Shape [2]

# p will be broadcast to shape [5, 7, 3] to match counts.
counts = [[...]]  # Shape [5, 7, 3]
dist.prob(counts)  # Shape [5, 7]
```

Create a 2-batch of 3-class distributions.

```python
p = [[.1, .2, .7], [.3, .3, .4]]  # Shape [2, 3]
dist = Multinomial(n=[4., 5], p=p)

counts = [[2., 1, 1], [3, 1, 1]]
dist.prob(counts)  # Shape [2]
```
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#### `tf.contrib.distributions.Multinomial.__init__(n, logits=None, p=None, validate_args=False, allow_nan_stats=True, name='Multinomial')` {#Multinomial.__init__}

Initialize a batch of Multinomial distributions.

##### Args:


*  <b>`n`</b>: Non-negative floating point tensor with shape broadcastable to
    `[N1,..., Nm]` with `m >= 0`. Defines this as a batch of
    `N1 x ... x Nm` different Multinomial distributions.  Its components
    should be equal to integer values.
*  <b>`logits`</b>: Floating point tensor representing the log-odds of a
    positive event with shape broadcastable to `[N1,..., Nm, k], m >= 0`,
    and the same dtype as `n`. Defines this as a batch of `N1 x ... x Nm`
    different `k` class Multinomial distributions. Only one of `logits` or
    `p` should be passed in.
*  <b>`p`</b>: Positive floating point tensor with shape broadcastable to
    `[N1,..., Nm, k]` `m >= 0` and same dtype as `n`.  Defines this as
    a batch of `N1 x ... x Nm` different `k` class Multinomial
    distributions. `p`'s components in the last portion of its shape should
    sum up to 1. Only one of `logits` or `p` should be passed in.
*  <b>`validate_args`</b>: `Boolean`, default `False`.  Whether to assert valid
    values for parameters `n` and `p`, and `x` in `prob` and `log_prob`.
    If `False`, correct behavior is not guaranteed.
*  <b>`allow_nan_stats`</b>: `Boolean`, default `True`.  If `False`, raise an
    exception if a statistic (e.g. mean/mode/etc...) is undefined for any
    batch member.  If `True`, batch members with valid parameters leading to
    undefined statistics will return NaN for this statistic.
*  <b>`name`</b>: The name to prefix Ops created by this distribution class.


*  <b>`Examples`</b>: 

```python
# Define 1-batch of 2-class multinomial distribution,
# also known as a Binomial distribution.
dist = Multinomial(n=2., p=[.1, .9])

# Define a 2-batch of 3-class distributions.
dist = Multinomial(n=[4., 5], p=[[.1, .3, .6], [.4, .05, .55]])
```


- - -

#### `tf.contrib.distributions.Multinomial.allow_nan_stats` {#Multinomial.allow_nan_stats}

Python boolean 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 boolean.


- - -

#### `tf.contrib.distributions.Multinomial.batch_shape(name='batch_shape')` {#Multinomial.batch_shape}

Shape of a single sample from a single event index as a 1-D `Tensor`.

The product of the dimensions of the `batch_shape` is the number of
independent distributions of this kind the instance represents.

##### Args:


*  <b>`name`</b>: name to give to the op

##### Returns:


*  <b>`batch_shape`</b>: `Tensor`.


- - -

#### `tf.contrib.distributions.Multinomial.cdf(value, name='cdf', **condition_kwargs)` {#Multinomial.cdf}

Cumulative distribution function.

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

```
cdf(x) := P[X <= x]
```

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`cdf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.Multinomial.copy(**override_parameters_kwargs)` {#Multinomial.copy}

Creates a deep copy of the distribution.

Note: the copy distribution may continue to depend on the original
intialization arguments.

##### Args:


*  <b>`**override_parameters_kwargs`</b>: String/value dictionary of initialization
    arguments to override with new values.

##### Returns:


*  <b>`distribution`</b>: A new instance of `type(self)` intitialized from the union
    of self.parameters and override_parameters_kwargs, i.e.,
    `dict(self.parameters, **override_parameters_kwargs)`.


- - -

#### `tf.contrib.distributions.Multinomial.dtype` {#Multinomial.dtype}

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


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#### `tf.contrib.distributions.Multinomial.entropy(name='entropy')` {#Multinomial.entropy}

Shannon entropy in nats.


- - -

#### `tf.contrib.distributions.Multinomial.event_shape(name='event_shape')` {#Multinomial.event_shape}

Shape of a single sample from a single batch as a 1-D int32 `Tensor`.

##### Args:


*  <b>`name`</b>: name to give to the op

##### Returns:


*  <b>`event_shape`</b>: `Tensor`.


- - -

#### `tf.contrib.distributions.Multinomial.get_batch_shape()` {#Multinomial.get_batch_shape}

Shape of a single sample from a single event index as a `TensorShape`.

Same meaning as `batch_shape`. May be only partially defined.

##### Returns:


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


- - -

#### `tf.contrib.distributions.Multinomial.get_event_shape()` {#Multinomial.get_event_shape}

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

Same meaning as `event_shape`. May be only partially defined.

##### Returns:


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


- - -

#### `tf.contrib.distributions.Multinomial.is_continuous` {#Multinomial.is_continuous}




- - -

#### `tf.contrib.distributions.Multinomial.is_reparameterized` {#Multinomial.is_reparameterized}




- - -

#### `tf.contrib.distributions.Multinomial.log_cdf(value, name='log_cdf', **condition_kwargs)` {#Multinomial.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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`logcdf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.Multinomial.log_pdf(value, name='log_pdf', **condition_kwargs)` {#Multinomial.log_pdf}

Log probability density function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`log_prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if not `is_continuous`.


- - -

#### `tf.contrib.distributions.Multinomial.log_pmf(value, name='log_pmf', **condition_kwargs)` {#Multinomial.log_pmf}

Log probability mass function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`log_pmf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if `is_continuous`.


- - -

#### `tf.contrib.distributions.Multinomial.log_prob(value, name='log_prob', **condition_kwargs)` {#Multinomial.log_prob}

Log probability density/mass function (depending on `is_continuous`).


Additional documentation from `Multinomial`:

For each batch of counts `[n_1,...,n_k]`, `P[counts]` is the probability
that after sampling `n` draws from this Multinomial distribution, the
number of draws falling in class `j` is `n_j`.  Note that different
sequences of draws can result in the same counts, thus the probability
includes a combinatorial coefficient.

Note that input "counts" must be a non-negative tensor with dtype `dtype`
and whose shape can be broadcast with `self.p` and `self.n`.  For fixed
leading dimensions, the last dimension represents counts for the
corresponding Multinomial distribution in `self.p`. `counts` is only legal
if it sums up to `n` and its components are equal to integer values.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`log_prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.Multinomial.log_survival_function(value, name='log_survival_function', **condition_kwargs)` {#Multinomial.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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:

  `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type
    `self.dtype`.


- - -

#### `tf.contrib.distributions.Multinomial.logits` {#Multinomial.logits}

Vector of coordinatewise logits.


- - -

#### `tf.contrib.distributions.Multinomial.mean(name='mean')` {#Multinomial.mean}

Mean.


- - -

#### `tf.contrib.distributions.Multinomial.mode(name='mode')` {#Multinomial.mode}

Mode.


- - -

#### `tf.contrib.distributions.Multinomial.n` {#Multinomial.n}

Number of trials.


- - -

#### `tf.contrib.distributions.Multinomial.name` {#Multinomial.name}

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


- - -

#### `tf.contrib.distributions.Multinomial.p` {#Multinomial.p}

Vector of probabilities summing to one.

Each element is the probability of drawing that coordinate.


- - -

#### `tf.contrib.distributions.Multinomial.param_shapes(cls, sample_shape, name='DistributionParamShapes')` {#Multinomial.param_shapes}

Shapes of parameters given the desired shape of a call to `sample()`.

Subclasses should override static method `_param_shapes`.

##### Args:


*  <b>`sample_shape`</b>: `Tensor` or python list/tuple. Desired shape of a call to
    `sample()`.
*  <b>`name`</b>: name to prepend ops with.

##### Returns:

  `dict` of parameter name to `Tensor` shapes.


- - -

#### `tf.contrib.distributions.Multinomial.param_static_shapes(cls, sample_shape)` {#Multinomial.param_static_shapes}

param_shapes with static (i.e. TensorShape) shapes.

##### Args:


*  <b>`sample_shape`</b>: `TensorShape` or python list/tuple. Desired shape of a call
    to `sample()`.

##### Returns:

  `dict` of parameter name to `TensorShape`.

##### Raises:


*  <b>`ValueError`</b>: if `sample_shape` is a `TensorShape` and is not fully defined.


- - -

#### `tf.contrib.distributions.Multinomial.parameters` {#Multinomial.parameters}

Dictionary of parameters used to instantiate this `Distribution`.


- - -

#### `tf.contrib.distributions.Multinomial.pdf(value, name='pdf', **condition_kwargs)` {#Multinomial.pdf}

Probability density function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if not `is_continuous`.


- - -

#### `tf.contrib.distributions.Multinomial.pmf(value, name='pmf', **condition_kwargs)` {#Multinomial.pmf}

Probability mass function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`pmf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if `is_continuous`.


- - -

#### `tf.contrib.distributions.Multinomial.prob(value, name='prob', **condition_kwargs)` {#Multinomial.prob}

Probability density/mass function (depending on `is_continuous`).


Additional documentation from `Multinomial`:

For each batch of counts `[n_1,...,n_k]`, `P[counts]` is the probability
that after sampling `n` draws from this Multinomial distribution, the
number of draws falling in class `j` is `n_j`.  Note that different
sequences of draws can result in the same counts, thus the probability
includes a combinatorial coefficient.

Note that input "counts" must be a non-negative tensor with dtype `dtype`
and whose shape can be broadcast with `self.p` and `self.n`.  For fixed
leading dimensions, the last dimension represents counts for the
corresponding Multinomial distribution in `self.p`. `counts` is only legal
if it sums up to `n` and its components are equal to integer values.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.Multinomial.sample(sample_shape=(), seed=None, name='sample', **condition_kwargs)` {#Multinomial.sample}

Generate samples of the specified shape.

Note that a call to `sample()` without arguments will generate a single
sample.

##### Args:


*  <b>`sample_shape`</b>: 0D or 1D `int32` `Tensor`. Shape of the generated samples.
*  <b>`seed`</b>: Python integer seed for RNG
*  <b>`name`</b>: name to give to the op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`samples`</b>: a `Tensor` with prepended dimensions `sample_shape`.


- - -

#### `tf.contrib.distributions.Multinomial.sample_n(n, seed=None, name='sample_n', **condition_kwargs)` {#Multinomial.sample_n}

Generate `n` samples.

##### Args:


*  <b>`n`</b>: `Scalar` `Tensor` of type `int32` or `int64`, the number of
    observations to sample.
*  <b>`seed`</b>: Python integer seed for RNG
*  <b>`name`</b>: name to give to the op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`samples`</b>: a `Tensor` with a prepended dimension (n,).

##### Raises:


*  <b>`TypeError`</b>: if `n` is not an integer type.


- - -

#### `tf.contrib.distributions.Multinomial.std(name='std')` {#Multinomial.std}

Standard deviation.


- - -

#### `tf.contrib.distributions.Multinomial.survival_function(value, name='survival_function', **condition_kwargs)` {#Multinomial.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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:

  Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type
    `self.dtype`.


- - -

#### `tf.contrib.distributions.Multinomial.validate_args` {#Multinomial.validate_args}

Python boolean indicated possibly expensive checks are enabled.


- - -

#### `tf.contrib.distributions.Multinomial.variance(name='variance')` {#Multinomial.variance}

Variance.


