tf.compat.v1.distributions.Gamma

Gamma distribution.

Inherits From: Distribution

tf.compat.v1.distributions.Gamma(
    concentration, rate, validate_args=False, allow_nan_stats=True, name='Gamma'
)

The Gamma distribution is defined over positive real numbers using parameters concentration (aka "alpha") and rate (aka "beta").

Mathematical Details

The probability density function (pdf) is,

pdf(x; alpha, beta, x > 0) = x**(alpha - 1) exp(-x beta) / Z
Z = Gamma(alpha) beta**(-alpha)

where:

concentration = alpha, alpha > 0,
rate = beta, beta > 0,
Z is the normalizing constant, and,
Gamma is the gamma function.

The cumulative density function (cdf) is,

cdf(x; alpha, beta, x > 0) = GammaInc(alpha, beta x) / Gamma(alpha)

where GammaInc is the lower incomplete Gamma function.

The parameters can be intuited via their relationship to mean and stddev,

concentration = alpha = (mean / stddev)**2
rate = beta = mean / stddev**2 = concentration / mean

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

dist = tfd.Gamma(concentration=3.0, rate=2.0)
dist2 = tfd.Gamma(concentration=[3.0, 4.0], rate=[2.0, 3.0])

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

concentration = tf.constant(3.0)
rate = tf.constant(2.0)
dist = tfd.Gamma(concentration, rate)
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, [concentration, rate])

Args
`concentration`	Floating point tensor, the concentration params of the distribution(s). Must contain only positive values.
`rate`	Floating point tensor, the inverse scale params of the distribution(s). Must contain only positive values.
`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.

Raises
`TypeError`	if `concentration` and `rate` are different dtypes.

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.
`concentration`	Concentration parameter.
`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.
`name`	Name prepended to all ops created by this `Distribution`.
`parameters`	Dictionary of parameters used to instantiate this `Distribution`.
`rate`	Rate parameter.
`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.

Returns
`is_scalar_event`	`bool` scalar `Tensor`.

`kl_divergence`

View source

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`	`tfp.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`

View source

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`

View source

log_prob(
    value, name='log_prob'
)

Log probability density/mass function.

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`

View source

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`

View source

mean(
    name='mean'
)

Mean.

`mode`

View source

mode(
    name='mode'
)

Mode.

Additional documentation from Gamma:

The mode of a gamma distribution is (shape - 1) / rate when shape > 1, and NaN otherwise. If self.allow_nan_stats is False, an exception will be raised rather than returning NaN.

`param_shapes`

View source

@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.

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`

View source

@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.

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`

View source

prob(
    value, name='prob'
)

Probability density/mass function.

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`

View source

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`

View source

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`

View source

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`

View source

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`

View source

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()`.

tf.compat.v1.distributions.Gamma

Mathematical Details

Examples

Args

Raises

Attributes

Methods

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

is_scalar_batch

is_scalar_event

kl_divergence

log_cdf

log_prob

log_survival_function

mean

mode

param_shapes

param_static_shapes

prob

quantile

sample

stddev

survival_function

variance

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`

`log_prob`

`log_survival_function`

`mean`

`mode`

`param_shapes`

`param_static_shapes`

`prob`

`quantile`

`sample`

`stddev`

`survival_function`

`variance`