tfp.distributions.Beta

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Beta distribution.

Inherits From: AutoCompositeTensorDistribution, Distribution, AutoCompositeTensor

tfp.distributions.Beta(
    concentration1,
    concentration0,
    validate_args=False,
    allow_nan_stats=True,
    force_probs_to_zero_outside_support=False,
    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 the paper

Michael Figurnov, Shakir Mohamed, Andriy Mnih. Implicit Reparameterization Gradients, 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]

# Define an equivalent Beta distribution parameterized by `mean` and
# `total_concentration`:
dist = tfd.Beta.experimental_from_mean_concentration(
  mean=0.5, total_concentration=alpha+beta)

# 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])

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
True
list(b.submodules) == [c]
True
list(c.submodules) == []
True

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.

cdf

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cdf(
    value, name='cdf', **kwargs
)

Cumulative distribution function.

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

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

Additional documentation from Beta:

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

covariance

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covariance(
    name='covariance', **kwargs
)

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.

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), (Shannon) 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 types with built-in registrations: Beta

entropy

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entropy(
    name='entropy', **kwargs
)

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.

experimental_default_event_space_bijector

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experimental_default_event_space_bijector(
    *args, **kwargs
)

Bijector mapping the reals (R**n) to the event space of the distribution.

Distributions with continuous support may implement _default_event_space_bijector which returns a subclass of tfp.bijectors.Bijector that maps R**n to the distribution's event space. For example, the default bijector for the Beta distribution is tfp.bijectors.Sigmoid(), which maps the real line to [0, 1], the support of the Beta distribution. The default bijector for the CholeskyLKJ distribution is tfp.bijectors.CorrelationCholesky, which maps R^(k * (k-1) // 2) to the submanifold of k x k lower triangular matrices with ones along the diagonal.

The purpose of experimental_default_event_space_bijector is to enable gradient descent in an unconstrained space for Variational Inference and Hamiltonian Monte Carlo methods. Some effort has been made to choose bijectors such that the tails of the distribution in the unconstrained space are between Gaussian and Exponential.

For distributions with discrete event space, or for which TFP currently lacks a suitable bijector, this function returns None.

experimental_fit

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@classmethod
experimental_fit(
    value, sample_ndims=1, validate_args=False, **init_kwargs
)

Instantiates a distribution that maximizes the likelihood of x.

experimental_from_mean_concentration

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@classmethod
experimental_from_mean_concentration(
    mean, total_concentration, **kwargs
)

Constructs a Beta from its mean and total concentration.

Experimental: Naming, location of this API may change.

Total concentration, sometimes called "sample size", is the sum of the two concentration parameters (concentration1 and concentration0 in __init__).

experimental_from_mean_variance

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@classmethod
experimental_from_mean_variance(
    mean, variance, **kwargs
)

Constructs a Beta from its mean and variance.

Experimental: Naming, location of this API may change.

Variance must be less than mean * (1. - mean), and in particular less than the maximal variance of 0.25, which occurs with mean = 0.5.

experimental_local_measure

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experimental_local_measure(
    value, backward_compat=False, **kwargs
)

Returns a log probability density together with a TangentSpace.

A TangentSpace allows us to calculate the correct push-forward density when we apply a transformation to a Distribution on a strict submanifold of R^n (typically via a Bijector in the TransformedDistribution subclass). The density correction uses the basis of the tangent space.

experimental_sample_and_log_prob

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experimental_sample_and_log_prob(
    sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)

Samples from this distribution and returns the log density of the sample.

The default implementation simply calls sample and log_prob:

def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
  x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
  return x, self.log_prob(x, **kwargs)

However, some subclasses may provide more efficient and/or numerically stable implementations.

is_scalar_batch

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is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

is_scalar_event

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is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

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 (Shannon) cross entropy, and H[.] denotes (Shannon) entropy.

other types with built-in registrations: Beta

log_cdf

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log_cdf(
    value, name='log_cdf', **kwargs
)

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:

log_prob

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log_prob(
    value, name='log_prob', **kwargs
)

Log probability density/mass function.

Additional documentation from Beta:

log_survival_function

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log_survival_function(
    value, name='log_survival_function', **kwargs
)

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.

mean

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mean(
    name='mean', **kwargs
)

Mean.

mode

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mode(
    name='mode', **kwargs
)

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(). (deprecated)

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.

param_static_shapes

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@classmethod
param_static_shapes(
    sample_shape
)

param_shapes with static (i.e. TensorShape) shapes. (deprecated)

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.

parameter_properties

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@classmethod
parameter_properties(
    dtype=tf.float32, num_classes=None
)

Returns a dict mapping constructor arg names to property annotations.

This dict should include an entry for each of the distribution's Tensor-valued constructor arguments.

Distribution subclasses are not required to implement _parameter_properties, so this method may raise NotImplementedError. Providing a _parameter_properties implementation enables several advanced features, including:

• Distribution batch slicing (sliced_distribution = distribution[i:j]).
• Automatic inference of _batch_shape and _batch_shape_tensor, which must otherwise be computed explicitly.
• Automatic instantiation of the distribution within TFP's internal property tests.
• Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.

In the future, parameter property annotations may enable additional functionality; for example, returning Distribution instances from tf.vectorized_map.

prob

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prob(
    value, name='prob', **kwargs
)

Probability density/mass function.

Additional documentation from Beta:

quantile

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quantile(
    value, name='quantile', **kwargs
)

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

sample

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sample(
    sample_shape=(), seed=None, name='sample', **kwargs
)

Generate samples of the specified shape.

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

stddev

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stddev(
    name='stddev', **kwargs
)

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.

survival_function

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survival_function(
    value, name='survival_function', **kwargs
)

Survival function.

Given random variable X, the survival function is defined:

survival_function(x) = P[X > x]
                     = 1 - P[X <= x]
                     = 1 - cdf(x).

unnormalized_log_prob

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unnormalized_log_prob(
    value, name='unnormalized_log_prob', **kwargs
)

Potentially unnormalized log probability density/mass function.

This function is similar to log_prob, but does not require that the return value be normalized. (Normalization here refers to the total integral of probability being one, as it should be by definition for any probability distribution.) This is useful, for example, for distributions where the normalization constant is difficult or expensive to compute. By default, this simply calls log_prob.

variance

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variance(
    name='variance', **kwargs
)

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.

with_name_scope

@classmethod
with_name_scope(
    method
)

Decorator to automatically enter the module name scope.

class MyModule(tf.Module):
  @tf.Module.with_name_scope
  def __call__(self, x):
    if not hasattr(self, 'w'):
      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
    return tf.matmul(x, self.w)

Using the above module would produce tf.Variables and tf.Tensors whose names included the module name:

mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

__getitem__

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__getitem__(
    slices
)

Slices the batch axes of this distribution, returning a new instance.

b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape  # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape  # => [3, 1, 5, 2, 4]

x = tf.random.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape  # => [4, 5, 3]
mvn.event_shape  # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => [2]

__iter__

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__iter__()