InverseGamma

class cuqi.distribution.InverseGamma(shape=None, location=None, scale=None, is_symmetric=False, **kwargs)

Multivariate inverse gamma distribution of independent random variables x_i. Each is distributed according to the PDF function

\[f(x) = (x-\beta)^{(-\alpha-1)} * \exp(-\gamma/(x-\beta)) / (\gamma^{(-\alpha)}*\Gamma(\alpha))\]

where shape \(\alpha\), location \(\beta\) and scale \(\gamma\) are the shape, location and scale of x_i, respectively. And \(\Gamma\) is the Gamma function.

Parameters:

• shape (float or array_like) – The shape parameter

• location (float or array_like) – The location of the inverse gamma distribution. The support of the pdf function is the Cartesian product of the open intervals (location_1, infinity), (location_2, infinity), …, (location_dim, infinity).

• scale (float or array_like) – The scale of the inverse gamma distribution (non-negative)

Example

# Generate an InverseGamma distribution
import numpy as np
import cuqi
import matplotlib.pyplot as plt
shape = [1,2]
location = 0
scale = 1
rng = np.random.RandomState(1)
x = cuqi.distribution.InverseGamma(shape, location, scale)
samples = x.sample(1000, rng=rng)
samples.hist_chain(0, bins=70)
plt.figure()
samples.hist_chain(1, bins=70)

__init__(shape=None, location=None, scale=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, location, scale, 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.