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tf.compat.v1.distributions.Multinomial

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Multinomial distribution.

Inherits From: Distribution

tf.compat.v1.distributions.Multinomial(
    total_count, logits=None, probs=None, validate_args=False, allow_nan_stats=True,
    name='Multinomial'
)

This Multinomial distribution is parameterized by probs, a (batch of) length-K prob (probability) vectors (K > 1) such that tf.reduce_sum(probs, -1) = 1, and a total_count number of trials, i.e., the number of trials per draw from the Multinomial. It is defined over a (batch of) length-K vector counts such that tf.reduce_sum(counts, -1) = total_count. The Multinomial is identically the Binomial distribution when K = 2.

Mathematical Details

The 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; pi, N) = prod_j (pi_j)**n_j / Z
Z = (prod_j n_j!) / N!

where:

  • probs = pi = [pi_0, ..., pi_{K-1}], pi_j > 0, sum_j pi_j = 1,
  • total_count = N, N a positive integer,
  • Z is the normalization constant, and,
  • N! denotes N factorial.

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

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

logits = [-50., -43, 0]
dist = Multinomial(total_count=4., logits=logits)

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

p = [.2, .3, .5]
dist = Multinomial(total_count=4., probs=p)

The distribution functions can be evaluated on counts.

# 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.

p = [[.1, .2, .7], [.3, .3, .4]]  # Shape [2, 3]
dist = Multinomial(total_count=[4., 5], probs=p)

counts = [[2., 1, 1], [3, 1, 1]]
dist.prob(counts)  # Shape [2]

dist.sample(5) # Shape [5, 2, 3]

total_count 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.
logits Floating point tensor representing unnormalized log-probabilities of a positive event with shape broadcastable to [N1,..., Nm, K] m >= 0, and the same dtype as total_count. Defines this as a batch of N1 x ... x Nm different K class Multinomial distributions. Only one of logits or probs should be passed in.
probs Positive floating point tensor with shape broadcastable to [N1,..., Nm, K] m >= 0 and same dtype as total_count. Defines this as a batch of N1 x ... x Nm different K class Multinomial distributions. probs's components in the last portion of its shape should sum to 1. Only one of logits or probs should be passed in.
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.

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.

dtype The DType of Tensors handled by this Distribution.
event_shape Shape of a single sample from a single batch as a TensorShape.

May be partially defined or unknown.

logits Vector of coordinatewise logits.
name Name prepended to all ops created by this Distribution.
parameters Dictionary of parameters used to instantiate this Distribution.
probs Probability of drawing a 1 in that coordinate.
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_count Number of trials used to construct a sample.
validate_args Python bool indicating possibly expensive checks are enabled.

Methods

batch_shape_tensor

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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

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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 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

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copy(
    **override_parameters_kwargs
)

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

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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.

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

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cross_entropy(
    other, name='cross_entropy'
)

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

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entropy(
    name='entropy'
)

Shannon entropy in nats.

event_shape_tensor

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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

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is_scalar_batch(
    name='is_scalar_batch'
)

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