tf.compat.v1.distributions.Categorical

Categorical distribution.

Inherits From: Distribution

tf.compat.v1.distributions.Categorical(
    logits=None, probs=None, dtype=tf.dtypes.int32, validate_args=False,
    allow_nan_stats=True, name='Categorical'
)

The Categorical distribution is parameterized by either probabilities or log-probabilities of a set of K classes. It is defined over the integers {0, 1, ..., K}.

The Categorical distribution is closely related to the OneHotCategorical and Multinomial distributions. The Categorical distribution can be intuited as generating samples according to argmax{ OneHotCategorical(probs) } itself being identical to argmax{ Multinomial(probs, total_count=1) }.

Mathematical Details

The probability mass function (pmf) is,

pmf(k; pi) = prod_j pi_j**[k == j]

Pitfalls

The number of classes, K, must not exceed:

the largest integer representable by self.dtype, i.e., 2**(mantissa_bits+1) (IEEE 754),
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

Creates a 3-class distribution with the 2nd class being most likely.

dist = Categorical(probs=[0.1, 0.5, 0.4])
n = 1e4
empirical_prob = tf.cast(
    tf.histogram_fixed_width(
      dist.sample(int(n)),
      [0., 2],
      nbins=3),
    dtype=tf.float32) / n
# ==> array([ 0.1005,  0.5037,  0.3958], dtype=float32)

Creates a 3-class distribution with the 2nd class being most likely. Parameterized by logits rather than probabilities.

dist = Categorical(logits=np.log([0.1, 0.5, 0.4])
n = 1e4
empirical_prob = tf.cast(
    tf.histogram_fixed_width(
      dist.sample(int(n)),
      [0., 2],
      nbins=3),
    dtype=tf.float32) / n
# ==> array([0.1045,  0.5047, 0.3908], dtype=float32)

Creates a 3-class distribution with the 3rd class being most likely. The distribution functions can be evaluated on counts.

# counts is a scalar.
p = [0.1, 0.4, 0.5]
dist = Categorical(probs=p)
dist.prob(0)  # Shape []

# p will be broadcast to [[0.1, 0.4, 0.5], [0.1, 0.4, 0.5]] to match counts.
counts = [1, 0]
dist.prob(counts)  # Shape [2]

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

Args
`logits`	An N-D `Tensor`, `N >= 1`, representing the log probabilities of a set of Categorical distributions. The first `N - 1` dimensions index into a batch of independent distributions and the last dimension represents a vector of logits for each class. Only one of `logits` or `probs` should be passed in.
`probs`	An N-D `Tensor`, `N >= 1`, representing the probabilities of a set of Categorical distributions. The first `N - 1` dimensions index into a batch of independent distributions and the last dimension represents a vector of probabilities for each class. Only one of `logits` or `probs` should be passed in.
`dtype`	The type of the event samples (default: int32).
`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.
`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.
`event_size`	Scalar `int32` tensor: the number of classes.
`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`	Vector of coordinatewise probabilities.
`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`.
`validate_args`	Python `bool` indicating possibly expensive checks are enabled.

Methods

`batch_shape_tensor`

View source

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`

View source

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`

View source

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`

View source

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`

View source

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`

View source

entropy(
    name='entropy'
)

Shannon entropy in nats.

`event_shape_tensor`

View source

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`

View source

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`

View source

is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

Args
`name`	Python `str` prepended to names of ops created by this function.

tf.compat.v1.distributions.Categorical

Mathematical Details

Pitfalls

Examples

Args

Attributes

Methods

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

is_scalar_batch

is_scalar_event

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`is_scalar_batch`

`is_scalar_event`