Gamma#

class cuqi.distribution.Gamma(shape=None, rate=None, is_symmetric=False, **kwargs)#

Represents a multivariate Gamma distribution characterized by shape and rate parameters of independent random variables x_i. Each is distributed according to the PDF function

\[f(x_i; \alpha, \beta) = \beta^\alpha x_i^{\alpha-1} \exp(-\beta x_i) / \Gamma(\alpha)\]

where shape \(\alpha\) and rate \(\beta\) are the parameters of the distribution, and \(\Gamma\) is the Gamma function.

In case shape and/or rate are arrays, the pdf looks like

\[f(x_i; \alpha_i, \beta_i) = \beta_i^{\alpha_i} x_i^{\alpha_i-1} \exp(-\beta_i x_i) / \Gamma(\alpha_i)\]

Parameters:

shape (float or array_like, optional) – The shape parameter of the Gamma distribution. Must be positive.
rate (float or array_like, optional) – The rate parameter of the Gamma distribution. Must be positive.

Examples

import numpy as np
import cuqi
import matplotlib.pyplot as plt

# Create a multivariate Gamma distribution with the same shape and rate parameters
shape = 1
rate = 1e-4
gamma_dist = cuqi.distribution.Gamma(shape=shape, rate=rate, geometry=10)

# Generate samples
samples = gamma_dist.sample(10000)

# Plot histogram of samples for index 0
samples.hist_chain(0, bins=70)

import numpy as np
import cuqi
import matplotlib.pyplot as plt

# Create a multivariate Gamma distribution with different shape and rate parameters
shape = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
rate = [1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1, 1e2, 1e3, 1e4, 1e5]
gamma_dist = cuqi.distribution.Gamma(shape=shape, rate=rate)

# Generate samples
samples = gamma_dist.sample(10000)

# Plot histogram of samples for index 0
samples.hist_chain(0, bins=70)

__init__(shape=None, rate=None, is_symmetric=False, **kwargs)#

Initialize the core properties of the distribution.

Parameters:

name (str, default None) – Name of distribution.
geometry (Geometry, default _DefaultGeometry (or None)) – Geometry of distribution.
is_symmetric (bool, default None) – Indicator if distribution is symmetric.

Methods

`__init__`([shape, rate, is_symmetric])	Initialize the core properties of the distribution.
`cdf`(x)
`disable_FD`()	Disable finite difference approximation for logd gradient.
`enable_FD`([epsilon])	Enable finite difference approximation for logd gradient.
`get_conditioning_variables`()	Return the conditioning variables of this distribution (if any).
`get_mutable_variables`()	Return any public variable that is mutable (attribute or property) except those in the ignore_vars list
`get_parameter_names`()	Returns the names of the parameters that the density can be evaluated at or conditioned on.
`gradient`(args, *kwargs)	Returns the gradient of the log density at x.
`logd`(args, *kwargs)	Evaluate the un-normalized log density function of the distribution.
`logpdf`(x)	Evaluate the log probability density function of the distribution.
`pdf`(x)	Evaluate the log probability density function of the distribution.
`sample`([N])	Sample from the distribution.
`to_likelihood`(data)	Convert conditional distribution to a likelihood function given observed data

Attributes

`FD_enabled`	Returns True if finite difference approximation of the logd gradient is enabled.
`FD_epsilon`	Spacing for the finite difference approximation of the logd gradient.
`dim`	Return the dimension of the distribution based on the geometry.
`geometry`	Return the geometry of the distribution.
`is_cond`	Returns True if instance (self) is a conditional distribution.
`name`	Name of the random variable associated with the density.
`rate`	Rate parameter of the Gamma distribution.
`scale`	Scale parameter of the Gamma distribution.
`shape`	Shape parameter of the Gamma distribution.