bayesm (version 1.1-2)

rnmixGibbs: Gibbs Sampler for Normal Mixtures

Description

rnmixGibbs implements a Gibbs Sampler for normal mixtures.

Usage

rnmixGibbs(Data, Prior, Mcmc)

Arguments

Data

list(y)

Prior

list(Mubar,A,nu,V,a,ncomp) (only ncomp required)

Mcmc

list(R,keep) (R required)

Value

• a list containing:
• probdraw
  R/keep x ncomp array of mixture prob draws
• zdraw
  R/keep x nobs array of indicators of mixture comp identity for each obs
• compdraw
  R/keep lists of lists of comp parm draws

concept

• bayes
• MCMC
• normal mixtures
• Gibbs Sampling

Details

Model: $y_i$ $\sim$ $N(mu_{ind_i},Sigma_{ind_i})$. ind $\sim$ iid multinomial(p). p is a ncomp x 1 vector of probs. Priors: $mu_j$ $\sim$ $N(mubar,Sigma_j (x) A^{-1})$. $mubar=vec(Mubar)$. $Sigma_j$ $\sim$ IW(nu,V). note: this is the natural conjugate prior -- a special case of multivariate regression. $p$ $\sim$ Dirchlet(a). Output of the components is in the form of a list of lists. compsdraw[[i]] is ith draw -- list of ncomp lists. compsdraw[[i]][[j]] is list of parms for jth normal component. jcomp=compsdraw[[i]][j]]. Then jth comp $\sim$ $N(jcomp[[1]],Sigma)$, $Sigma$ = t(R)%*%R, $R^{-1}$ = jcomp[[2]]. List arguments contain:

• y

{ n x k array of data (rows are obs) } Mubar{ k x 1 array with prior mean of normal comp means (def: 0)} A{ 1 x 1 precision parameter for prior on mean of normal comp (def: .01)} nu{ d.f. parameter for prior on Sigma (normal comp cov matrix) (def: k+3)} V{ k x k location matrix of IW prior on Sigma (def: nuI)} a{ ncomp x 1 vector of Dirichlet prior parms (def: rep(5,ncomp))} ncomp{ number of normal components to be included } R{ number of MCMC draws } keep{ MCMC thinning parm: keep every keepth draw (def: 1)}

References

For further discussion, see Bayesian Statistics and Marketing by Allenby, McCulloch, and Rossi, Chapter 5. http://gsbwww.uchicago.edu/fac/peter.rossi/research/bsm.html

See Also

rmixture, rmixGibbs ,eMixMargDen, momMix

Examples

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

set.seed(66)
dim=5;  k=3   # dimension of simulated data and number of "true" components
sigma = matrix(rep(0.5,dim^2),nrow=dim);diag(sigma)=1
sigfac = c(1,1,1);mufac=c(1,2,3); compsmv=list()
for(i in 1:k) compsmv[[i]] = list(mu=mufac[i]*1:dim,sigma=sigfac[i]*sigma)
comps = list() # change to "rooti" scale
for(i in 1:k) comps[[i]] = list(mu=compsmv[[i]][[1]],rooti=solve(chol(compsmv[[i]][[2]])))
pvec=(1:k)/sum(1:k)

nobs=5000
dm = rmixture(nobs,pvec,comps)

Data=list(y=dm$x)
ncomp=9
Prior=list(ncomp=ncomp,a=c(rep(1,ncomp)))
Mcmc=list(R=R,keep=1)
out=rnmixGibbs(Data=Data,Prior=Prior,Mcmc=Mcmc)

tmom=momMix(matrix(pvec,nrow=1),list(comps))
if(R < 1000) {begin=1} else {begin=500}
pmom=momMix(out$probdraw[begin:R,],out$compdraw[begin:R])
mat=rbind(tmom$mu,pmom$mu)
rownames(mat)=c("true","post expect")
cat("mu and posterior expectation of mu",fill=TRUE)
print(mat)
mat=rbind(tmom$sd,pmom$sd)
rownames(mat)=c("true","post expect")
cat("std dev and posterior expectation of sd",fill=TRUE)
print(mat)
mat=rbind(as.vector(tmom$corr),as.vector(pmom$corr))
rownames(mat)=c("true","post expect")
cat("corr and posterior expectation of corr",fill=TRUE)
print(t(mat))