tf.compat.v1.distributions.DirichletMultinomial

Dirichlet-Multinomial compound distribution.

Inherits From: 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 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])

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]

total_countNon-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.
concentrationPositive 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_argsPython 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_statsPython 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.
namePython str name prefixed to Ops created by this class.

allow_nan_statsPython 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_shapeShape 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.

concentrationConcentration parameter; expected prior counts for that coordinate.
dtypeThe DType of Tensors handled by this Distribution.
event_shapeShape of a single sample from a single batch as a TensorShape.

May be partially defined or unknown.

nameName prepended to all ops created by this Distribution.
parametersDictionary of parameters used to instantiate this Distribution.
reparameterization_typeDescribes how samples from the distribution are reparameterized.

Currently this is one of the static instances distributions.FULLY_REPARAMETERIZED or distributions.NOT_REPARAMETERIZED.

total_concentrationSum of last dim of concentration parameter.
total_countNumber of trials used to construct a sample.
validate_argsPython bool indicating possibly expensive checks are enabled.

Methods

batch_shape_tensor

View source

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
namename to give to the op

Returns
batch_shapeTensor.

cdf

View source

Cumulative distribution function.

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

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

Args
valuefloat or double Tensor.
namePython str prepended to names of ops created by this function.

Returns
cdfa 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_kwargsString/value dictionary of initialization arguments to override with new values.

Returns
distributionA 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.

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
namePython str prepended to names of ops created by this function.

Returns
covarianceFloating-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

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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
othertfp.distributions.Distribution instance.
namePython str prepended to names of ops created by this function.

Returns
cross_entropyself.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shanon) cross entropy.

entropy

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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
namename to give to the op

Returns
event_shapeTensor.

is_scalar_batch

View source

Indicates that batch_shape == [].

Args
namePython str prepended to names of ops created by this function.

Returns
is_scalar_batchbool scalar Tensor.

is_scalar_event

View source

Indicates that event_shape == [].

Args
namePython str prepended to names of ops created by this function.

Returns
is_scalar_eventbool scalar Tensor.

kl_divergence

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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
othertfp.distributions.Distribution instance.
namePython str prepended to names of ops created by this function.

Returns
kl_divergenceself.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

log_cdf

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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
valuefloat or double Tensor.
namePython str prepended to names of ops created by this function.

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

log_prob

View source

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.

Args
valuefloat or double Tensor.
namePython str prepended to names of ops created by this function.

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

log_survival_function

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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
valuefloat or double Tensor.
namePython 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

View source

Mean.

mode

View source

Mode.

param_shapes

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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_shapeTensor or python list/tuple. Desired shape of a call to sample().
namename to prepend ops with.

Returns
dict of parameter name to Tensor shapes.

param_static_shapes

View source

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_shapeTensorShape or python list/tuple. Desired shape of a call to sample().

Returns
dict of parameter name to TensorShape.

Raises
ValueErrorif sample_shape is a TensorShape and is not fully defined.

prob

View source

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.

Args
valuefloat or double Tensor.
namePython str prepended to names of ops created by this function.

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

quantile

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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
valuefloat or double Tensor.
namePython str prepended to names of ops created by this function.

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

sample

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Generate samples of the specified shape.

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

Args
sample_shape0D or 1D int32 Tensor. Shape of the generated samples.
seedPython integer seed for RNG
namename to give to the op.

Returns
samplesa Tensor with prepended dimensions sample_shape.

stddev

View source

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
namePython str prepended to names of ops created by this function.

Returns
stddevFloating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

survival_function

View source

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
valuefloat or double Tensor.
namePython 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

View source

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
namePython str prepended to names of ops created by this function.

Returns
varianceFloating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().