Cauchy

class cuqi.distribution.Cauchy(location=None, scale=None, is_symmetric=True, **kwargs)

Multivariate Cauchy distribution of independent random variables.

Each is distributed according to the PDF function

\[\frac{1}{\pi\gamma(1+\frac{(x-\mu)^2}{\gamma^2})}\]

where \(\mu\) is the location parameter and \(\gamma\) is the scale parameter.

Parameters:

• location (float or array_like) – Location parameter

• scale (float or array_like) – Scale parameter

Example

# % Generate a Cauchy distribution
import cuqi
location = 0
scale  = 1
x = cuqi.distribution.Cauchy(location, scale)
x.logpdf(0) # -1.1447298858494002

# % Generate a multivariate Cauchy distribution
import cuqi
location = [0, 1]
scale  = [1, 3]
x = cuqi.distribution.Cauchy(location, scale)
x.logpdf([0, 0]) # -3.4934325760247367

__init__(location=None, scale=None, is_symmetric=True, **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__([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(x)	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.
location	Location parameter
name	Name of the random variable associated with the density.
scale	Scale parameter