rmnpGibbs {bayesm} R Documentation

## Gibbs Sampler for Multinomial Probit

### Description

rmnpGibbs implements the McCulloch/Rossi Gibbs Sampler for the multinomial probit model.

### Usage

rmnpGibbs(Data, Prior, Mcmc)

### Arguments

 Data list(y, X, p) Prior list(betabar, A, nu, V) Mcmc list(R, keep, nprint, beta0, sigma0)

### Details

#### Model and Priors

w_i = X_i\beta + e with e \sim N(0, \Sigma). Note: w_i and e are (p-1) x 1.
y_i = j if w_{ij} > max(0,w_{i,-j}) for j=1, \ldots, p-1 and where w_{i,-j} means elements of w_i other than the jth.
y_i = p, if all w_i < 0

\beta is not identified. However, \beta/sqrt(\sigma_{11}) and \Sigma/\sigma_{11} are identified. See reference or example below for details.

\beta \sim N(betabar,A^{-1})
\Sigma \sim IW(nu,V)

#### Argument Details

Data = list(y, X, p)

 y:  n x 1 vector of multinomial outcomes (1, ..., p) X:  n*(p-1) x k design matrix. To make X matrix use createX with DIFF=TRUE p:  number of alternatives

Prior = list(betabar, A, nu, V) [optional]

 betabar:  k x 1 prior mean (def: 0) A:  k x k prior precision matrix (def: 0.01*I) nu:  d.f. parameter for Inverted Wishart prior (def: (p-1)+3) V:  PDS location parameter for Inverted Wishart prior (def: nu*I)

Mcmc = list(R, keep, nprint, beta0, sigma0) [only R required]

 R:  number of MCMC draws keep:  MCMC thinning parameter -- keep every keepth draw (def: 1) nprint:  print the estimated time remaining for every nprint'th draw (def: 100, set to 0 for no print) beta0:  initial value for beta (def: 0) sigma0:  initial value for sigma (def: I)

### Value

A list containing:

 betadraw R/keep x k matrix of betadraws sigmadraw R/keep x (p-1)*(p-1) matrix of sigma draws – each row is the vector form of sigma

### Author(s)

Peter Rossi, Anderson School, UCLA, perossichi@gmail.com.

### References

For further discussion, see Chapter 4, Bayesian Statistics and Marketing by Rossi, Allenby, and McCulloch.

rmvpGibbs

### Examples

if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=2000} else {R=10}
set.seed(66)

simmnp = function(X, p, n, beta, sigma) {
indmax = function(x) {which(max(x)==x)}
Xbeta = X%*%beta
w = as.vector(crossprod(chol(sigma),matrix(rnorm((p-1)*n),ncol=n))) + Xbeta
w = matrix(w, ncol=(p-1), byrow=TRUE)
maxw = apply(w, 1, max)
y = apply(w, 1, indmax)
y = ifelse(maxw < 0, p, y)
return(list(y=y, X=X, beta=beta, sigma=sigma))
}

p = 3
n = 500
beta = c(-1,1,1,2)
Sigma = matrix(c(1, 0.5, 0.5, 1), ncol=2)
k = length(beta)
X1 = matrix(runif(n*p,min=0,max=2),ncol=p)
X2 = matrix(runif(n*p,min=0,max=2),ncol=p)
X = createX(p, na=2, nd=NULL, Xa=cbind(X1,X2), Xd=NULL, DIFF=TRUE, base=p)

simout = simmnp(X,p,500,beta,Sigma)

Data1 = list(p=p, y=simout$y, X=simout$X)
Mcmc1 = list(R=R, keep=1)

out = rmnpGibbs(Data=Data1, Mcmc=Mcmc1)

cat(" Summary of Betadraws ", fill=TRUE)
betatilde = out$betadraw / sqrt(out$sigmadraw[,1])
attributes(betatilde)$class = "bayesm.mat" summary(betatilde, tvalues=beta) cat(" Summary of Sigmadraws ", fill=TRUE) sigmadraw = out$sigmadraw / out$sigmadraw[,1] attributes(sigmadraw)$class = "bayesm.var"