LaplacesDemon (version 16.1.6)

dist.Normal.Wishart: Normal-Wishart Distribution

Description

These functions provide the density and random number generation for the normal-Wishart distribution.

Usage

dnormwishart(mu, mu0, lambda, Omega, S, nu, log=FALSE) 
rnormwishart(n=1, mu0, lambda, S, nu)

Arguments

mu

This is data or parameters in the form of a vector of length \(k\) or a matrix with \(k\) columns.

mu0

This is mean vector \(\mu_0\) with length \(k\) or matrix with \(k\) columns.

lambda

This is a positive-only scalar.

n

This is the number of random draws.

nu

This is the scalar degrees of freedom \(\nu\).

Omega

This is a \(k \times k\) precision matrix \(\Omega\).

S

This is the symmetric, positive-semidefinite, \(k \times k\) scale matrix \(\textbf{S}\).

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dnormwishart gives the density and rnormwishart generates random deviates and returns a list with two components.

Details

• Application: Continuous Multivariate

• Density: \(p(\mu, \Omega) = \mathcal{N}(\mu | \mu_0, (\lambda\Omega)^{-1}) \mathcal{W}(\Omega | \nu, \textbf{S})\)

• Inventors: Unknown

• Notation 1: \((\mu, \Omega) \sim \mathcal{NW}(\mu_0, \lambda, \textbf{S}, \nu)\)

• Notation 2: \(p(\mu, \Omega) = \mathcal{NW}(\mu, \Omega | \mu_0, \lambda, \textbf{S}, \nu)\)

• Parameter 1: location vector \(\mu_0\)

• Parameter 2: \(\lambda > 0\)

• Parameter 3: symmetric, positive-semidefinite \(k \times k\) scale matrix \(\textbf{S}\)

• Parameter 4: degrees of freedom \(\nu \ge k\)

• Mean: Unknown

• Variance: Unknown

• Mode: Unknown

The normal-Wishart distribution, or Gaussian-Wishart distribution, is a multivariate four-parameter continuous probability distribution. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix.

See Also

dmvnp and dwishart.

Examples

# NOT RUN {
library(LaplacesDemon)
K <- 3
mu <- rnorm(K)
mu0 <- rnorm(K)
nu <- K + 1
S <- diag(K)
lambda <- runif(1) #Real scalar
Omega <- as.positive.definite(matrix(rnorm(K^2),K,K))
x <- dnormwishart(mu, mu0, lambda, Omega, S, nu, log=TRUE)
out <- rnormwishart(n=10, mu0, lambda, S, nu)
joint.density.plot(out$mu[,1], out$mu[,2], color=TRUE)
# }