tf.compat.v1.distributions.DirichletMultinomial

Dirichlet-Multinomial compound distribution.

Inherits From: Distribution

tf.compat.v1.distributions.DirichletMultinomial(
    total_count, concentration, validate_args=False, allow_nan_stats=True,
    name='DirichletMultinomial'
)

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.

Choose class probabilities: probs = [p_0,...,p_{K-1}] ~ Dir(concentration)
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 counts corresponds single Dirichlet-Multinomial distributions.

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

Pitfalls

The number of classes, K, must not exceed:

the largest integer representable by self.dtype, i.e., 2**(mantissa_bits+1) (IEE754),
the maximum Tensor index, i.e., 2**31-1.

In other words,

K <= min(2**31-1, {
  tf.float16: 2**11,
  tf.float32: 2**24,
  tf.float64: 2**53 }[param.dtype])

Note: This condition is validated only when self.validate_args = True.

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]

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.

Attributes
`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.
`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.
`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.
`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`.
`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.

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

Args
`other`	`tfp.distributions.Distribution` instance.
`name`	Python `str` prepended to names of ops created by this function.

Args
`sample_shape`	`Tensor` or python list/tuple. Desired shape of a call to `sample()`.
`name`	name to prepend ops with.

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.

tf.compat.v1.distributions.DirichletMultinomial

Mathematical Details

Pitfalls

Examples

Methods

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

`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`