tf.distributions.Dirichlet

Class Dirichlet

Inherits From: Distribution

Aliases:

• Class tf.contrib.distributions.Dirichlet
• Class tf.distributions.Dirichlet

Defined in tensorflow/python/ops/distributions/dirichlet.py.

See the guide: Statistical Distributions (contrib) > Multivariate distributions

Dirichlet distribution.

The Dirichlet distribution is defined over the (k-1)-simplex using a positive, length-k vector concentration (k > 1). The Dirichlet is identically the Beta distribution when k = 2.

Mathematical Details

The Dirichlet is a distribution over the open (k-1)-simplex, i.e.,

S^{k-1} = { (x_0, ..., x_{k-1}) in R^k : sum_j x_j = 1 and all_j x_j > 0 }.

The probability density function (pdf) is,

pdf(x; alpha) = prod_j x_j**(alpha_j - 1) / Z
Z = prod_j Gamma(alpha_j) / Gamma(sum_j alpha_j)

where:

• x in S^{k-1}, i.e., the (k-1)-simplex,
• concentration = alpha = [alpha_0, ..., alpha_{k-1}], alpha_j > 0,
• Z is the normalization constant aka the multivariate beta function, and,
• Gamma is the gamma function.

The concentration represents mean total counts of class occurrence, i.e.,

concentration = alpha = mean * total_concentration

where mean in S^{k-1} and total_concentration is a positive real number representing a mean total count.

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

Examples

# Create a single trivariate Dirichlet, with the 3rd class being three times
# more frequent than the first. I.e., batch_shape=[], event_shape=[3].
alpha = [1., 2, 3]
dist = Dirichlet(alpha)

dist.sample([4, 5])  # shape: [4, 5, 3]

# x has one sample, one batch, three classes:
x = [.2, .3, .5]   # shape: [3]
dist.prob(x)       # shape: []

# x has two samples from one batch:
x = [[.1, .4, .5],
     [.2, .3, .5]]
dist.prob(x)         # shape: [2]

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

# Create batch_shape=[2], event_shape=[3]:
alpha = [[1., 2, 3],
         [4, 5, 6]]   # shape: [2, 3]
dist = Dirichlet(alpha)

dist.sample([4, 5])  # shape: [4, 5, 2, 3]

x = [.2, .3, .5]
# x will be broadcast as [[.2, .3, .5],
#                         [.2, .3, .5]],
# thus matching batch_shape [2, 3].
dist.prob(x)         # 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 counts for that coordinate.

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.

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.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

__init__

__init__(
    concentration,
    validate_args=False,
    allow_nan_stats=True,
    name='Dirichlet'
)

Initialize a batch of Dirichlet distributions.

Args:

• concentration: Positive floating-point Tensor indicating mean number of class occurrences; aka "alpha". Implies self.dtype, and self.batch_shape, self.event_shape, i.e., if concentration.shape = [N1, N2, ..., Nm, k] then batch_shape = [N1, N2, ..., Nm] and event_shape = [k].
• 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: 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

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.

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

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: tf.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

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: Python str prepended to names of ops created by this function.

Returns:

• is_scalar_batch: bool scalar Tensor.

is_scalar_event

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

• name: Python str prepended to names of ops created by this function.

Returns:

• is_scalar_event: bool scalar Tensor.

kl_divergence

kl_divergence(
    other,
    name='kl_divergence'
)

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:

• other: tf.distributions.Distribution instance.
• name: Python str prepended to names of ops created by this function.

Returns:

• kl_divergence: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

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: Python str prepended to names of ops created by this function.

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 Dirichlet:

Note: value must be a non-negative tensor with dtype self.dtype and be in the (self.event_shape() - 1)-simplex, i.e., tf.reduce_sum(value, -1) = 1. It must have a shape compatible with self.batch_shape() + self.event_shape().

Args:

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

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: Python 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

mean(name='mean')

Mean.

mode

mode(name='mode')

Mode.

Additional documentation from Dirichlet:

Note: The mode is undefined when any concentration <= 1. If self.allow_nan_stats is True, NaN is used for undefined modes. If self.allow_nan_stats is False an exception is raised when one or more modes are undefined.

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 Dirichlet:

Note: value must be a non-negative tensor with dtype self.dtype and be in the (self.event_shape() - 1)-simplex, i.e., tf.reduce_sum(value, -1) = 1. It must have a shape compatible with self.batch_shape() + self.event_shape().

Args:

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

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: Python str prepended to names of ops created by this function.

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: Python str prepended to names of ops created by this function.

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: Python 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

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: Python str prepended to names of ops created by this function.

Returns:

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