| lmn_post {LMN} | R Documentation |
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)
Arguments
suff |
An object of class |
prior |
A list with elements |
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}.
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.
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