# tf.contrib.distributions.DirichletMultinomial

### `class tf.contrib.distributions.DirichletMultinomial`

Dirichlet-Multinomial compound distribution.

The Dirichlet-Multinomial distribution is parameterized by a (batch of) length-`k` `concentration` vectors (`k > 1`) and a `total_count` number of trials, i.e., the number of trials per draw from the DirichletMultinomial. It is defined over a (batch of) length-`k` vector `counts` such that `tf.reduce_sum(counts, -1) = total_count`. The Dirichlet-Multinomial is identically the Beta-Binomial distribution when `k = 2`.

#### Mathematical Details

The Dirichlet-Multinomial is a distribution over `k`-class counts, i.e., a length-`k` vector of non-negative integer `counts = n = [n_0, ..., n_{k-1}]`.

The probability mass function (pmf) is,

```pmf(n; alpha, N) = Beta(alpha + n) / (prod_j n_j!) / Z
Z = Beta(alpha) / N!
```

where:

• `concentration = alpha = [alpha_0, ..., alpha_{k-1}]`, `alpha_j > 0`,
• `total_count = N`, `N` a positive integer,
• `N!` is `N` factorial, and,
• `Beta(x) = prod_j Gamma(x_j) / Gamma(sum_j x_j)` is the multivariate beta function, and,
• `Gamma` is the gamma function.

Dirichlet-Multinomial is a compound distribution, i.e., its samples are generated as follows.

1. Choose class probabilities: `probs = [p_0,...,p_{k-1}] ~ Dir(concentration)`
2. Draw integers: `counts = [n_0,...,n_{k-1}] ~ Multinomial(total_count, probs)`

The last `concentration` dimension parametrizes a single Dirichlet-Multinomial distribution. When calling distribution functions (e.g., `dist.prob(counts)`), `concentration`, `total_count` and `counts` are broadcast to the same shape. The last dimension of of `counts` corresponds single Dirichlet-Multinomial distributions.

Distribution parameters are automatically broadcast in all functions; see examples for details.

#### Examples

```alpha = [1, 2, 3]
n = 2
dist = DirichletMultinomial(n, alpha)
```

Creates a 3-class distribution, with the 3rd class is most likely to be drawn. The distribution functions can be evaluated on counts.

```# counts same shape as alpha.
counts = [0, 0, 2]
dist.prob(counts)  # Shape []

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

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

Creates a 2-batch of 3-class distributions.

```alpha = [[1, 2, 3], [4, 5, 6]]  # Shape [2, 3]
n = [3, 3]
dist = DirichletMultinomial(n, alpha)

# counts will be broadcast to [[2, 1, 0], [2, 1, 0]] to match alpha.
counts = [2, 1, 0]
dist.prob(counts)  # 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.

### `concentration`

Concentration parameter; expected prior counts for that coordinate.

### `dtype`

The `DType` of `Tensor`s 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.

### `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`.

### `total_concentration`

Sum of last dim of concentration parameter.

### `total_count`

Number of trials used to construct a sample.

### `validate_args`

Python `bool` indicating possibly expensive checks are enabled.

## Methods

### `__init__`

```__init__(
total_count,
concentration,
validate_args=False,
allow_nan_stats=True,
name='DirichletMultinomial'
)
```

Initialize a batch of DirichletMultinomial distributions.

#### Args:

• `total_count`: Non-negative floating point tensor, whose dtype is the same as `concentration`. The shape is broadcastable to `[N1,..., Nm]` with `m >= 0`. Defines this as a batch of `N1 x ... x Nm` different Dirichlet multinomial distributions. Its components should be equal to integer values.
• `concentration`: Positive floating point tensor, whose dtype is the same as `n` with shape broadcastable to `[N1,..., Nm, k]` `m >= 0`. Defines this as a batch of `N1 x ... x Nm` different `k` class Dirichlet multinomial distributions.
• `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.

### `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`: The name to give this op.

#### 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.

Additional documentation from `DirichletMultinomial`:

The covariance for each batch member is defined as the following:

```Var(X_j) = n * alpha_j / alpha_0 * (1 - alpha_j / alpha_0) *
(n + alpha_0) / (1 + alpha_0)
```

where `concentration = alpha` and `total_concentration = alpha_0 = sum_j alpha_j`.

The covariance between elements in a batch is defined as:

```Cov(X_i, X_j) = -n * alpha_i * alpha_j / alpha_0 ** 2 *
(n + alpha_0) / (1 + alpha_0)
```

#### Args:

• `name`: The name to give this op.

#### 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)`.

### `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`: The name to give this op.

#### Returns:

• `is_scalar_batch`: `bool` scalar `Tensor`.

### `is_scalar_event`

```is_scalar_event(name='is_scalar_event')
```

Indicates that `event_shape == []`.

#### Args:

• `name`: The name to give this op.

#### Returns:

• `is_scalar_event`: `bool` scalar `Tensor`.

### `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`: The name to give this op.

#### 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 `DirichletMultinomial`:

For each batch of counts, `value = [n_0, ..., n_{k-1}]`, `P[value]` is the probability that after sampling `self.total_count` draws from this Dirichlet-Multinomial distribution, the number of draws falling in class `j` is `n_j`. Since this definition is exchangeable; different sequences have the same counts so the probability includes a combinatorial coefficient.

Note: `value` must be a non-negative tensor with dtype `self.dtype`, have no fractional components, and such that `tf.reduce_sum(value, -1) = self.total_count`. Its shape must be broadcastable with `self.concentration` and `self.total_count`.

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: The name to give this op.

#### 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`: The name to give this op.

#### 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 `DirichletMultinomial`:

For each batch of counts, `value = [n_0, ..., n_{k-1}]`, `P[value]` is the probability that after sampling `self.total_count` draws from this Dirichlet-Multinomial distribution, the number of draws falling in class `j` is `n_j`. Since this definition is exchangeable; different sequences have the same counts so the probability includes a combinatorial coefficient.

Note: `value` must be a non-negative tensor with dtype `self.dtype`, have no fractional components, and such that `tf.reduce_sum(value, -1) = self.total_count`. Its shape must be broadcastable with `self.concentration` and `self.total_count`.

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: The name to give this op.

#### 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`: The name to give this op.

#### 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`: The name to give this op.

#### 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`: The name to give this op.

#### 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`: The name to give this op.

#### Returns:

• `variance`: Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.

https://www.tensorflow.org/api_docs/python/tf/contrib/distributions/DirichletMultinomial

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