# dist.Multivariate.Cauchy.Precision: Multivariate Cauchy Distribution: Precision Parameterization In LaplacesDemonR/LaplacesDemon: Complete Environment for Bayesian Inference

## Description

These functions provide the density and random number generation for the multivariate Cauchy distribution. These functions use the precision parameterization.

## Usage

 1 2 dmvcp(x, mu, Omega, log=FALSE) rmvcp(n=1, mu, Omega)

## Arguments

 x This is either a vector of length k or a matrix with a number of columns, k, equal to the number of columns in precision matrix Omega. n This is the number of random draws. mu This is a numeric vector representing the location parameter, mu (the mean vector), of the multivariate distribution. It must be of length k, as defined above. Omega This is a k x k positive-definite precision matrix Omega. log Logical. If log=TRUE, then the logarithm of the density is returned.

## Details

• Application: Continuous Multivariate

• Density:

p(theta) = (Gamma((nu+k)/2) / (Gamma(1/2)*1^(k/2)*pi^(k/2))) * |Omega|^(1/2) * (1 + (theta-mu)^T Omega (theta-mu))^(-(1+k)/2)

• Inventor: Unknown (to me, anyway)

• Notation 1: theta ~ MC[k](mu, Omega^(-1))

• Notation 2: p(theta) = MC[k](theta | mu, Omega^(-1))

• Parameter 1: location vector mu

• Parameter 2: positive-definite k x k precision matrix Omega

• Mean: E(theta) = mu

• Variance: var(theta) = undefined

• Mode: mode(theta) = mu

The multivariate Cauchy distribution is a multidimensional extension of the one-dimensional or univariate Cauchy distribution. A random vector is considered to be multivariate Cauchy-distributed if every linear combination of its components has a univariate Cauchy distribution. The multivariate Cauchy distribution is equivalent to a multivariate t distribution with 1 degree of freedom.

The Cauchy distribution is known as a pathological distribution because its mean and variance are undefined, and it does not satisfy the central limit theorem.

It is usually parameterized with mean and a covariance matrix, or in Bayesian inference, with mean and a precision matrix, where the precision matrix is the matrix inverse of the covariance matrix. These functions provide the precision parameterization for convenience and familiarity. It is easier to calculate a multivariate Cauchy density with the precision parameterization, because a matrix inversion can be avoided.

This distribution has a mean parameter vector mu of length k, and a k x k precision matrix Omega, which must be positive-definite.

## Value

dmvcp gives the density and rmvcp generates random deviates.

## Author(s)

Statisticat, LLC. [email protected]