tf.compat.v1.distributions.Beta

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Beta distribution.

Inherits From: Distribution

tf.compat.v1.distributions.Beta(
    concentration1=None,
    concentration0=None,
    validate_args=False,
    allow_nan_stats=True,
    name='Beta'
)

The Beta distribution is defined over the (0, 1) interval using parameters concentration1 (aka "alpha") and concentration0 (aka "beta").

Mathematical Details

The probability density function (pdf) is,

pdf(x; alpha, beta) = x**(alpha - 1) (1 - x)**(beta - 1) / Z
Z = Gamma(alpha) Gamma(beta) / Gamma(alpha + beta)

where:

• concentration1 = alpha,
• concentration0 = beta,
• Z is the normalization constant, and,
• Gamma is the gamma function.

The concentration parameters represent mean total counts of a 1 or a 0, i.e.,

concentration1 = alpha = mean * total_concentration
concentration0 = beta  = (1. - mean) * total_concentration

where mean in (0, 1) and total_concentration is a positive real number representing a mean total_count = concentration1 + concentration0.

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

Samples of this distribution are reparameterized (pathwise differentiable). The derivatives are computed using the approach described in (Figurnov et al., 2018).

Examples

import tensorflow_probability as tfp
tfd = tfp.distributions

# Create a batch of three Beta distributions.
alpha = [1, 2, 3]
beta = [1, 2, 3]
dist = tfd.Beta(alpha, beta)

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

# `x` has three batch entries, each with two samples.
x = [[.1, .4, .5],
     [.2, .3, .5]]
# Calculate the probability of each pair of samples under the corresponding
# distribution in `dist`.
dist.prob(x)         # Shape [2, 3]

# Create batch_shape=[2, 3] via parameter broadcast:
alpha = [[1.], [2]]      # Shape [2, 1]
beta = [3., 4, 5]        # Shape [3]
dist = tfd.Beta(alpha, beta)

# alpha broadcast as: [[1., 1, 1,],
#                      [2, 2, 2]]
# beta broadcast as:  [[3., 4, 5],
#                      [3, 4, 5]]
# batch_Shape [2, 3]
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, 3]

Compute the gradients of samples w.r.t. the parameters:

alpha = tf.constant(1.0)
beta = tf.constant(2.0)
dist = tfd.Beta(alpha, beta)
samples = dist.sample(5)  # Shape [5]
loss = tf.reduce_mean(tf.square(samples))  # Arbitrary loss function
# Unbiased stochastic gradients of the loss function
grads = tf.gradients(loss, [alpha, beta])

concentration1 Positive floating-point Tensor indicating mean number of successes; aka "alpha". Implies self.dtype and self.batch_shape, i.e., concentration1.shape = [N1, N2, ..., Nm] = self.batch_shape. concentration0 Positive floating-point Tensor indicating mean number of failures; aka "beta". Otherwise has same semantics as concentration1. 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. concentration0 Concentration parameter associated with a 0 outcome. concentration1 Concentration parameter associated with a 1 outcome. 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. 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 concentration parameters. 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.

name name to give to the op

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]

Additional documentation from Beta:

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

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.

**override_parameters_kwargs String/value dictionary of initialization arguments to override with new values.

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.

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

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.

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

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.

name name to give to the op

event_shape Tensor.

is_scalar_batch

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is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

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

is_scalar_batch bool scalar Tensor.

is_scalar_event

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is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

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

is_scalar_event bool scalar Tensor.

kl_divergence

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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.

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

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

log_cdf

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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.

Additional documentation from Beta:

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

logcdf a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_prob

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log_prob(
    value, name='log_prob'
)

Log probability density/mass function.

Additional documentation from Beta:

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

log_prob a 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(
    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.

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

mean

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mean(
    name='mean'
)

Mean.

mode

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mode(
    name='mode'
)

Mode.

Additional documentation from Beta:

param_shapes

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@classmethod
param_shapes(
    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.

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

param_static_shapes

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@classmethod
param_static_shapes(
    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.

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

ValueError if sample_shape is a TensorShape and is not fully defined.

prob

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prob(
    value, name='prob'
)

Probability density/mass function.

Additional documentation from Beta:

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

prob a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

quantile

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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

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

quantile a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

sample

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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.

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.

samples a Tensor with prepended dimensions sample_shape.

stddev

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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.

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

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

survival_function

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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).

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

variance

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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.

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

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