LaplacesDemon (version 16.1.1)

dist.Normal.Inverse.Wishart: Normal-Inverse-Wishart Distribution

Description

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

Usage

dnorminvwishart(mu, mu0, lambda, Sigma, S, nu, log=FALSE) 
rnorminvwishart(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 \(\nu\).

Sigma

This is a \(k \times k\) covariance matrix \(\Sigma\).

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

dnorminvwishart gives the density and rnorminvwishart generates random deviates and returns a list with two components.

Details

• Application: Continuous Multivariate

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

• Inventors: Unknown

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

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

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

See Also

dmvn and dinvwishart.

Examples

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