rnmixGibbs {bayesm}  R Documentation 
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 keep th 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 loglikelihood (def: FALSE )

nmix
Details
nmix
is a list with 3 components. Several functions in the bayesm
package that involve a Dirichlet Process or mixtureofnormals 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 innermost 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 

nmix 
a list containing: 
Note
In this model, the component normal parameters are notidentified due to labelswitching. However, the fitted mixture of normals density is identified as it is invariant to labelswitching. 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, perossichi@gmail.com.
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)}