# dist.Multivariate.Cauchy.Precision.Cholesky: Multivariate Cauchy Distribution: Precision-Cholesky... 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 and Cholesky parameterization.

## Usage

 ```1 2``` ```dmvcpc(x, mu, U, log=FALSE) rmvcpc(n=1, mu, U) ```

## 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. `U` This is the k x k upper-triangular matrix that is Cholesky factor U of the 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) =

• 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. The precision matrix is replaced with the upper-triangular Cholesky factor, as in `chol`.

In practice, U is fully unconstrained for proposals when its diagonal is log-transformed. The diagonal is exponentiated after a proposal and before other calculations. Overall, Cholesky parameterization is faster than the traditional parameterization. Compared with `dmvcp`, `dmvcpc` must additionally matrix-multiply the Cholesky back to the covariance matrix, but it does not have to check for or correct the precision matrix to positive-definiteness, which overall is slower. Compared with `rmvcp`, `rmvcpc` is faster because the Cholesky decomposition has already been performed.

## Value

`dmvcpc` gives the density and `rmvcpc` generates random deviates.

## Author(s)

Statisticat, LLC. [email protected]

`chol`, `dcauchy`, `dmvcc`, `dmvtc`, `dmvtpc`, and `dwishartc`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```library(LaplacesDemon) x <- seq(-2,4,length=21) y <- 2*x+10 z <- x+cos(y) mu <- c(1,12,2) Omega <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) U <- chol(Omega) f <- dmvcpc(cbind(x,y,z), mu, U) X <- rmvcpc(1000, rep(0,2), diag(2)) X <- X[rowSums((X >= quantile(X, probs=0.025)) & (X <= quantile(X, probs=0.975)))==2,] joint.density.plot(X[,1], X[,2], color=TRUE) ```