LMN (version 1.1.3)

lmn_post: Parameters of the posterior conditional distribution of an LMN model.

Description

Calculates the parameters of the LMN model's Matrix-Normal Inverse-Wishart (MNIW) conjugate posterior distribution (see Details).

Usage

lmn_post(suff, prior)

Value

A list with elements named as in prior specifying the parameters of the posterior MNIW distribution. Elements Omega = NA and nu = NA specify that parameters Beta = 0 and Sigma = diag(q), respectively, are known and not to be estimated.

Arguments

suff

An object of class lmn_suff (see lmn_suff()).

prior

A list with elements Lambda, Omega, Psi, nu as returned by lmn_prior().

Details

The Matrix-Normal Inverse-Wishart (MNIW) distribution \((\boldsymbol{B}, \boldsymbol{\Sigma}) \sim \textrm{MNIW}(\boldsymbol{\Lambda}, \boldsymbol{\Omega}, \boldsymbol{\Psi}, \nu)\) on random matrices \(\boldsymbol{X}_{p \times q}\) and symmetric positive-definite \(\boldsymbol{\Sigma}_{q \times q}\) is defined as $$ \begin{array}{rcl} \boldsymbol{\Sigma} & \sim & \textrm{Inverse-Wishart}(\boldsymbol{\Psi}, \nu) \\ \boldsymbol{B} \mid \boldsymbol{\Sigma} & \sim & \textrm{Matrix-Normal}(\boldsymbol{\Lambda}, \boldsymbol{\Omega}^{-1}, \boldsymbol{\Sigma}), \end{array} $$ where the Matrix-Normal distribution is defined in lmn_suff().

The posterior MNIW distribution is required to be a proper distribution, but the prior is not. For example, prior = NULL corresponds to the noninformative prior $$ \pi(B, \boldsymbol{\Sigma}) \sim |\boldsymbol{Sigma}|^{-(q+1)/2}. $$

Examples

# generate data
n <- 50
q <- 2
p <- 3
Y <- matrix(rnorm(n*q),n,q) # response matrix
X <- matrix(rnorm(n*p),n,p) # covariate matrix
V <- .5 * exp(-(1:n)/n) # Toeplitz variance specification

suff <- lmn_suff(Y = Y, X = X, V = V, Vtype = "acf") # sufficient statistics