tfp.distributions.GammaGamma

Class `GammaGamma`

Gamma-Gamma distribution.

Inherits From: Distribution

Defined in python/distributions/gamma_gamma.py.

Gamma-Gamma is a compound distribution defined over positive real numbers using parameters concentration, mixing_concentration and mixing_rate.

This distribution is also referred to as the beta of the second kind (B2), and can be useful for transaction value modeling, as [(Fader and Hardi, 2013)][1].

Mathematical Details

It is derived from the following Gamma-Gamma hierarchical model by integrating out the random variable beta.

    beta ~ Gamma(alpha0, beta0)
X | beta ~ Gamma(alpha, beta)

where * concentration = alpha * mixing_concentration = alpha0 * mixing_rate = beta0

The probability density function (pdf) is

                                       x**(alpha - 1)
pdf(x; alpha, alpha0, beta0) = ---------------------------------
                               Z * (x + beta0)**(alpha + alpha0)

where the normalizing constant Z = Beta(alpha, alpha0) * beta0**(-alpha0).

Samples of this distribution are reparameterized as samples of the Gamma distribution are reparameterized using the technique described in [(Figurnov et al., 2018)][2].

References

[1]: Peter S. Fader, Bruce G. S. Hardi. The Gamma-Gamma Model of Monetary Value. Technical Report, 2013. http://www.brucehardie.com/notes/025/gamma_gamma.pdf

[2]: Michael Figurnov, Shakir Mohamed, Andriy Mnih. Implicit Reparameterization Gradients. arXiv preprint arXiv:1805.08498, 2018. https://arxiv.org/abs/1805.08498

`init`

__init__(
    concentration,
    mixing_concentration,
    mixing_rate,
    validate_args=False,
    allow_nan_stats=True,
    name='GammaGamma'
)

Initializes a batch of Gamma-Gamma distributions.

The parameters concentration and rate must be shaped in a way that supports broadcasting (e.g. concentration + mixing_concentration + mixing_rate is a valid operation).

Args:

concentration: Floating point tensor, the concentration params of the distribution(s). Must contain only positive values.
mixing_concentration: Floating point tensor, the concentration params of the mixing Gamma distribution(s). Must contain only positive values.
mixing_rate: Floating point tensor, the rate params of the mixing Gamma 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.

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.

`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.

`mixing_concentration`

Concentration parameter for the mixing Gamma distribution.

`mixing_rate`

Rate parameter for the mixing Gamma distribution.

`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 tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

Returns:

An instance of ReparameterizationType.

`validate_args`

Python bool indicating possibly expensive checks are enabled.

Methods

`getitem`

__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.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]

Args:

slices: slices from the [] operator

Returns:

dist: A new tfd.Distribution instance with sliced parameters.

`iter`

__iter__()

`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',
    **kwargs
)

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.
**kwargs: Named arguments forwarded to subclass implementation.

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',
    **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.

Args:

name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

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), (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.

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 (Shannon) cross entropy.

`entropy`

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

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

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.

Args:

value: float or double Tensor.
name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

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',
    **kwargs
)

Log probability density/mass function.

Args:

value: float or double Tensor.
name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

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',
    **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.

Args:

value: float or double Tensor.
name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

`mean`

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

Mean.

Additional documentation from GammaGamma:

The mean of a Gamma-Gamma distribution is concentration * mixing_rate / (mixing_concentration - 1), when mixing_concentration > 1, and NaN otherwise. If self.allow_nan_stats is False, an exception will be raised rather than returning NaN

`mode`

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

Mode.

`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',
    **kwargs
)

Probability density/mass function.

Args:

value: float or double Tensor.
name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

`quantile`

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

Args:

value: float or double Tensor.
name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

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',
    **kwargs
)

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.
**kwargs: Named arguments forwarded to subclass implementation.

Returns:

samples: a Tensor with prepended dimensions sample_shape.

`stddev`

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.

Args:

name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

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',
    **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).

Args:

value: float or double Tensor.
name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

`variance`

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.

Additional documentation from GammaGamma:

The variance of a Gamma-Gamma distribution is concentration**2 * mixing_rate**2 / ((mixing_concentration - 1)**2 * (mixing_concentration - 2)), when mixing_concentration > 2, and NaN otherwise. If self.allow_nan_stats is False, an exception will be raised rather than returning NaN

Args:

name: Python str prepended to names of ops created by this function.
**kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

tfp.distributions.GammaGamma

Class GammaGamma

Mathematical Details

References

__init__

Args:

Raises:

Properties

allow_nan_stats

Returns:

batch_shape

Returns:

concentration

dtype

event_shape

Returns:

mixing_concentration

mixing_rate

name

parameters

reparameterization_type

Returns:

validate_args

Methods

__getitem__

Args:

Returns:

__iter__

batch_shape_tensor

Args:

Returns:

cdf

Args:

Returns:

copy

Args:

Returns:

covariance

Args:

Returns:

cross_entropy

Args:

Returns:

entropy

event_shape_tensor

Args:

Returns:

is_scalar_batch

Args:

Returns:

is_scalar_event

Args:

Returns:

kl_divergence

Args:

Returns:

log_cdf

Args:

Returns:

log_prob

Args:

Returns:

log_survival_function

Args:

Returns:

mean

mode

param_shapes

Args:

Returns:

param_static_shapes

Args:

Returns:

Raises:

prob

Args:

Returns:

quantile

Args:

Returns:

Class `GammaGamma`

`init`

`allow_nan_stats`

`batch_shape`

`concentration`

`dtype`

`event_shape`

`mixing_concentration`

`mixing_rate`

`name`

`parameters`

`reparameterization_type`

`validate_args`

`getitem`

`iter`

`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`