rhierMnlDP {bayesm} | R Documentation |
rhierMnlDP
is a MCMC algorithm for a hierarchical multinomial logit with a Dirichlet Process Prior for the distribution of heteorogeneity. A base normal model is used so that the DP can be interpreted as allowing for a mixture of normals with as many components as there are panel units. This is a hybrid Gibbs Sampler with a RW Metropolis step for the MNL
coefficients for each panel unit. This procedure can be interpreted as a Bayesian semi-parameteric method in the sense that the DP prior can accomodate heterogeniety of an unknown form.
rhierMnlDP(Data, Prior, Mcmc)
Data |
list(p,lgtdata,Z) ( Z is optional) |
Prior |
list(deltabar,Ad,Prioralpha,lambda_hyper) (all are optional) |
Mcmc |
list(s,w,R,keep) (R required) |
Model:
y_i ~ MNL(X_i,beta_i). i=1,..., length(lgtdata). theta_i is nvar x 1.
beta_i= ZDelta[i,] + u_i.
Note: here ZDelta refers to Z%*%D, ZDelta[i,] is ith row of this product.
Delta is an nz x nvar array.
beta_i ~ N(mu_i,Sigma_i).
Priors:
theta_i=(mu_i,Sigma_i) ~ DP(G_0(lambda),alpha)
G_0(lambda):
mu_i | Sigma_i ~ N(0,Sigma_i (x) a^{-1})
Sigma_i ~ IW(nu,nu*v*I)
lambda(a,nu,v):
a ~ uniform[alim[1],alimb[2]]
nu ~ dim(data)-1 + exp(z)
z ~ uniform[dim(data)-1+nulim[1],nulim[2]]
v ~ uniform[vlim[1],vlim[2]]
alpha ~ (1-(alpha-alphamin)/(alphamax-alphamin))^power
alpha= alphamin then expected number of components = Istarmin
alpha= alphamax then expected number of components = Istarmax
Lists contain:
Data:
p
lgtdata
lgtdata[[i]]$y
lgtdata[[i]]$X
Prior:
deltabar
Ad
Prioralpha:
Istarmin
Istarmax
power
lambda_hyper:
alim
nulim
vlim
Mcmc:
R
keep
maxuniq
gridsize
a list containing:
Deltadraw |
R/keep x nz*nvar matrix of draws of Delta, first row is initial value |
betadraw |
nlgt x nvar x R/keep array of draws of betas |
nmix |
list of 3 components, probdraw, NULL, compdraw |
adraw |
R/keep draws of hyperparm a |
vdraw |
R/keep draws of hyperparm v |
nudraw |
R/keep draws of hyperparm nu |
Istardraw |
R/keep draws of number of unique components |
alphadraw |
R/keep draws of number of DP tightness parameter |
loglike |
R/keep draws of log-likelihood |
As is well known, Bayesian density estimation involves computing the predictive distribution of a "new" unit parameter,
theta_{n+1} (here "n"=nlgt). This is done by averaging the normal base distribution over draws from the distribution of theta_{n+1} given theta_1, ..., theta_n,alpha,lambda,Data.
To facilitate this, we store those draws from the predictive distribution of theta_{n+1} in a list structure compatible with other bayesm
routines that implement a finite mixture of normals.
More on nmix list:
contains the draws from the predictive distribution of a "new" observations parameters. These are simply the parameters of one normal distribution. We enforce compatibility with a mixture of k components in order to utilize generic summary
plotting functions.
Therefore,probdraw
is a vector of ones. zdraw
(indicator draws) is omitted as it is not necessary for density estimation. compdraw
contains the draws of the theta_{n+1} as a list of list of lists.
More on compdraw
component of return value list:
We parameterize the prior on Sigma_i such that mode(Sigma)= nu/(nu+2) vI. The support of nu enforces a non-degenerate IW density; nulim[1] > 0.
The default choices of alim,nulim, and vlim determine the location and approximate size of candidate "atoms" or possible normal components. The defaults are sensible given a reasonable scaling of the X variables. You want to insure that alim is set for a wide enough range of values (remember a is a precision parameter) and the v is big enough to propose Sigma matrices wide enough to cover the data range.
A careful analyst should look at the posterior distribution of a, nu, v to make sure that the support is set correctly in alim, nulim, vlim. In other words, if we see the posterior bunched up at one end of these support ranges, we should widen the range and rerun.
If you want to force the procedure to use many small atoms, then set nulim to consider only large values and set vlim to consider only small scaling constants. Set alphamax to a large number. This will create a very "lumpy" density estimate somewhat like the classical Kernel density estimates. Of course, this is not advised if you have a prior belief that densities are relatively smooth.
Note: Z should not include an intercept and is centered for ease of interpretation.
Large R values may be required (>20,000).
Peter Rossi, Graduate School of Business, University of Chicago, Peter.Rossi@ChicagoGsb.edu.
For further discussion, see Bayesian Statistics and Marketing
by Rossi, Allenby and McCulloch, Chapter 5.
http://faculty.chicagogsb.edu/peter.rossi/research/bsm.html
## if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=20000} else {R=10} set.seed(66) p=3 # num of choice alterns ncoef=3 nlgt=300 # num of cross sectional units nz=2 Z=matrix(runif(nz*nlgt),ncol=nz) Z=t(t(Z)-apply(Z,2,mean)) # demean Z ncomp=3 # no of mixture components Delta=matrix(c(1,0,1,0,1,2),ncol=2) comps=NULL comps[[1]]=list(mu=c(0,-1,-2),rooti=diag(rep(2,3))) comps[[2]]=list(mu=c(0,-1,-2)*2,rooti=diag(rep(2,3))) comps[[3]]=list(mu=c(0,-1,-2)*4,rooti=diag(rep(2,3))) pvec=c(.4,.2,.4) simmnlwX= function(n,X,beta) { ## simulate from MNL model conditional on X matrix k=length(beta) Xbeta=X%*%beta j=nrow(Xbeta)/n Xbeta=matrix(Xbeta,byrow=TRUE,ncol=j) Prob=exp(Xbeta) iota=c(rep(1,j)) denom=Prob%*%iota Prob=Prob/as.vector(denom) y=vector("double",n) ind=1:j for (i in 1:n) {yvec=rmultinom(1,1,Prob[i,]); y[i]=ind%*%yvec} return(list(y=y,X=X,beta=beta,prob=Prob)) } ## simulate data with a mixture of 3 normals simlgtdata=NULL ni=rep(50,300) for (i in 1:nlgt) { betai=Delta%*%Z[i,]+as.vector(rmixture(1,pvec,comps)$x) Xa=matrix(runif(ni[i]*p,min=-1.5,max=0),ncol=p) X=createX(p,na=1,nd=NULL,Xa=Xa,Xd=NULL,base=1) outa=simmnlwX(ni[i],X,betai) simlgtdata[[i]]=list(y=outa$y,X=X,beta=betai) } ## plot betas if(1){ ## set if(1) above to produce plots bmat=matrix(0,nlgt,ncoef) for(i in 1:nlgt) {bmat[i,]=simlgtdata[[i]]$beta} par(mfrow=c(ncoef,1)) for(i in 1:ncoef) hist(bmat[,i],breaks=30,col="magenta") } ## set Data and Mcmc lists keep=5 Mcmc1=list(R=R,keep=keep) Data1=list(p=p,lgtdata=simlgtdata,Z=Z) out=rhierMnlDP(Data=Data1,Mcmc=Mcmc1) cat("Summary of Delta draws",fill=TRUE) summary(out$Deltadraw,tvalues=as.vector(Delta)) if(0) { ## plotting examples plot(out$betadraw) plot(out$nmix) }