rnmixGibbs {bayesm}

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)

Mcmc

list(R, keep, nprint, Loglike)

Details

Model and Priors

y_i \sim N(\mu_{ind_i}, \Sigma_{ind_i})
ind \sim iid multinomial(p) with p an ncomp x 1 vector of probs

\mu_j \sim N(mubar, \Sigma_j (x) A^{-1}) with 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)

Argument Details

Data = list(y)

y:	n x k matrix of data (rows are obs)

Prior = list(Mubar, A, nu, V, a, ncomp) [only ncomp required]

Mubar:	1 x k vector with prior mean of normal component means (def: 0)
A:	1 x 1 precision parameter for prior on mean of normal component (def: 0.01)
nu:	d.f. parameter for prior on Sigma (normal component cov matrix) (def: k+3)
V:	k x k location matrix of IW prior on Sigma (def: nu*I)
a:	ncomp x 1 vector of Dirichlet prior parameters (def: rep(5,ncomp))
ncomp:	number of normal components to be included

Mcmc = list(R, keep, nprint, Loglike) [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)
LogLike:	logical flag for whether to compute the log-likelihood (def: FALSE)

nmix Details

nmix is a list with 3 components. Several functions in the bayesm package that involve a Dirichlet Process or mixture-of-normals return nmix. Across these functions, a common structure is used for nmix in order to utilize generic summary and plotting functions.

probdraw:	ncomp x R/keep matrix that reports the probability that each draw came from a particular component
zdraw:	R/keep x nobs matrix that indicates which component each draw is assigned to
compdraw:	A list of R/keep lists of ncomp lists. Each of the inner-most lists has 2 elemens: a vector of draws for mu and a matrix of draws for the Cholesky root of Sigma.

Value

A list containing:

ll

R/keep x 1 vector of log-likelihood values

nmix

a list containing: probdraw, zdraw, compdraw (see “nmix Details” section)

Note

In this model, the component normal parameters are not-identified due to label-switching. However, the fitted mixture of normals density is identified as it is invariant to label-switching. See chapter 5 of Rossi et al below for details.

Use eMixMargDen or momMix to compute posterior expectation or distribution of various identified parameters.

Author(s)

Peter Rossi, Anderson School, UCLA, [e-mail].

References

For further discussion, see Chapter 3, Bayesian Statistics and Marketing by Rossi, Allenby, and McCulloch.

See Also

rmixture, rmixGibbs ,eMixMargDen, momMix, mixDen, mixDenBi

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)

# change to "rooti" scale
comps = list() 
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 = 500
dm = rmixture(nobs, pvec, comps)

Data1 = list(y=dm$x)
ncomp = 9
Prior1 = list(ncomp=ncomp)
Mcmc1 = list(R=R, keep=1)

out = rnmixGibbs(Data=Data1, Prior=Prior1, Mcmc=Mcmc1)

cat("Summary of Normal Mixture Distribution", fill=TRUE)
summary(out$nmix)

tmom = momMix(matrix(pvec,nrow=1), list(comps))
mat = rbind(tmom$mu, tmom$sd)
cat(" True Mean/Std Dev", fill=TRUE)
print(mat)

## plotting examples
if(0){plot(out$nmix,Data=dm$x)}