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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,Zis the normalization constant, and,Gammais 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])
References:
Implicit Reparameterization Gradients: Figurnov et al., 2018 (pdf)
Args | |
|---|---|
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.
|
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. |
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
|
total_concentration
|
Sum of concentration parameters. |
validate_args
|
Python bool indicating possibly expensive checks are enabled.
|
Methods
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]
Additional documentation from Beta:
| 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.
| 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
|
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
entropy(
name='entropy'
)
Shannon entropy in nats.