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\name{rbprobitGibbs}
\alias{rbprobitGibbs}
\concept{bayes}
\concept{MCMC}
\concept{probit}
\concept{Gibbs Sampling}
\title{Gibbs Sampler (Albert and Chib) for Binary Probit}
\description{
\code{rbprobitGibbs} implements the Albert and Chib Gibbs Sampler for the binary probit model.
}
\usage{rbprobitGibbs(Data, Prior, Mcmc)}
\arguments{
\item{Data }{list(y, X)}
\item{Prior}{list(betabar, A)}
\item{Mcmc }{list(R, keep, nprint)}
}
\details{
\subsection{Model and Priors}{
\eqn{z = X\beta + e} with \eqn{e} \eqn{\sim}{~} \eqn{N(0, I)} \cr
\eqn{y = 1} if \eqn{z > 0}
\eqn{\beta} \eqn{\sim}{~} \eqn{N(betabar, A^{-1})}
}
\subsection{Argument Details}{
\emph{\code{Data = list(y, X)}}
\tabular{ll}{
\code{y: } \tab \eqn{n x 1} vector of 0/1 outcomes \cr
\code{X: } \tab \eqn{n x k} design matrix
}
\emph{\code{Prior = list(betabar, A)} [optional]}
\tabular{ll}{
\code{betabar: } \tab \eqn{k x 1} prior mean (def: 0) \cr
\code{A: } \tab \eqn{k x k} prior precision matrix (def: 0.01*I)
}
\emph{\code{Mcmc = list(R, keep, nprint)} [only \code{R} required]}
\tabular{ll}{
\code{R: } \tab number of MCMC draws \cr
\code{keep: } \tab MCMC thinning parameter -- keep every \code{keep}th draw (def: 1) \cr
\code{nprint: } \tab print the estimated time remaining for every \code{nprint}'th draw (def: 100, set to 0 for no print)
}
}
}
\value{
A list containing:
\item{betadraw }{ \eqn{R/keep x k} matrix of betadraws}
}
\author{Peter Rossi, Anderson School, UCLA, \email{perossichi@gmail.com}.}
\references{For further discussion, see Chapter 3, \emph{Bayesian Statistics and Marketing} by Rossi, Allenby, and McCulloch.}
\seealso{ \code{\link{rmnpGibbs}} }
\examples{
if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=2000} else {R=10}
set.seed(66)
## function to simulate from binary probit including x variable
simbprobit = function(X, beta) {
y = ifelse((X\%*\%beta + rnorm(nrow(X)))<0, 0, 1)
list(X=X, y=y, beta=beta)
}
nobs = 200
X = cbind(rep(1,nobs), runif(nobs), runif(nobs))
beta = c(0,1,-1)
nvar = ncol(X)
simout = simbprobit(X, beta)
Data1 = list(X=simout$X, y=simout$y)
Mcmc1 = list(R=R, keep=1)
out = rbprobitGibbs(Data=Data1, Mcmc=Mcmc1)
summary(out$betadraw, tvalues=beta)
## plotting example
if(0){plot(out$betadraw, tvalues=beta)}
}
\keyword{models}
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