rDPGibbs {bayesm}R Documentation

Density Estimation with Dirichlet Process Prior and Normal Base

Description

rDPGibbs implements a Gibbs Sampler to draw from the posterior for a normal mixture problem with a Dirichlet Process prior. A natural conjugate base prior is used along with priors on the hyper parameters of this distribution. One interpretation of this model is as a normal mixture with a random number of components that can grow with the sample size.

Usage

rDPGibbs(Prior, Data, Mcmc)

Arguments

Prior list(Prioralpha,lambda_hyper)
Data list(y)
Mcmc list(R,keep,maxuniq,SCALE,gridsize)

Details

Model:
y_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 on grid[alim[1],alimb[2]]
nu ~ uniform on grid[dim(data)-1 + exp(nulim[1]),dim(data)-1 +exp(nulim[2])]
v ~ uniform on grid[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

list arguments

Data:

Prioralpha:

lambda_hyper:

Mcmc:

output:
the basic output are draws from the predictive distribution of the data in the object, nmix. The average of these draws is the Bayesian analogue of a density estimate.

nmix:

Output of the components is in the form of a list of lists.
compdraw[[i]] is ith draw – list of lists.
compdraw[[i]][[1]] is list of parms for a draw from predictive.
compdraw[[i]][1]][[1]] is the mean vector. compdraw[[i]][[1]][[2]] is the inverse of Cholesky root. Sigma = t(R)%*%R, R^{-1} = compdraw[[i]][[1]][[2]].

Value

nmix a list containing: probdraw,zdraw,compdraw
alphadraw vector of draws of DP process tightness parameter
nudraw vector of draws of base prior hyperparameter
adraw vector of draws of base prior hyperparameter
vdraw vector of draws of base prior hyperparameter

Note

we parameterize the prior on Sigma_i such that mode(Sigma)= nu/(nu+2) vI. The support of nu enforces valid IW density; nulim[1] > 0

We use the structure for nmix that is compatible with the bayesm routines for finite mixtures of normals. This allows us to use the same summary and plotting methods.

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 that we scale the data. Without scaling, 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 Istarmax 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.

Author(s)

Peter Rossi, Graduate School of Business, University of Chicago, Peter.Rossi@ChicagoGsb.edu.

See Also

rnmixGibbs,rmixture, rmixGibbs , eMixMargDen, momMix, mixDen, mixDenBi

Examples

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

## simulate univariate data from Chi-Sq

set.seed(66)
N=200
chisqdf=8; y1=as.matrix(rchisq(N,df=chisqdf))

## set arguments for rDPGibbs

Data1=list(y=y1)
Prioralpha=list(Istarmin=1,Istarmax=10,power=.8)
Prior1=list(Prioralpha=Prioralpha)

Mcmc=list(R=R,keep=1,maxuniq=200)

out1=rDPGibbs(Prior=Prior1,Data=Data1,Mcmc)

if(0){
## plotting examples
rgi=c(0,20); grid=matrix(seq(from=rgi[1],to=rgi[2],length.out=50),ncol=1)
deltax=(rgi[2]-rgi[1])/nrow(grid)
plot(out1$nmix,Grid=grid,Data=y1)
## plot true density with historgram
plot(range(grid[,1]),1.5*range(dchisq(grid[,1],df=chisqdf)),type="n",xlab=paste("Chisq ; ",N," obs",sep=""), ylab="")
hist(y1,xlim=rgi,freq=FALSE,col="yellow",breaks=20,add=TRUE)
lines(grid[,1],dchisq(grid[,1],df=chisqdf)/(sum(dchisq(grid[,1],df=chisqdf))*deltax),col="blue",lwd=2)
}

## simulate bivariate data from the  "Banana" distribution (Meng and Barnard) 
banana=function(A,B,C1,C2,N,keep=10,init=10)
{ R=init*keep+N*keep
   x1=x2=0
   bimat=matrix(double(2*N),ncol=2)
  for (r in 1:R)
  { x1=rnorm(1,mean=(B*x2+C1)/(A*(x2^2)+1),sd=sqrt(1/(A*(x2^2)+1)))
  x2=rnorm(1,mean=(B*x2+C2)/(A*(x1^2)+1),sd=sqrt(1/(A*(x1^2)+1)))
  if (r>init*keep && r%%keep==0) {mkeep=r/keep; bimat[mkeep-init,]=c(x1,x2)} }
return(bimat)
}

set.seed(66)
nvar2=2
A=0.5; B=0; C1=C2=3
y2=banana(A=A,B=B,C1=C1,C2=C2,1000)

Data2=list(y=y2)
Prioralpha=list(Istarmin=1,Istarmax=10,power=.8)
Prior2=list(Prioralpha=Prioralpha)
Mcmc=list(R=R,keep=1,maxuniq=200)

out2=rDPGibbs(Prior=Prior2,Data=Data2,Mcmc)

if(0){
## plotting examples

rx1=range(y2[,1]); rx2=range(y2[,2])
x1=seq(from=rx1[1],to=rx1[2],length.out=50)
x2=seq(from=rx2[1],to=rx2[2],length.out=50)
grid=cbind(x1,x2)

plot(out2$nmix,Grid=grid,Data=y2)

## plot true bivariate density
tden=matrix(double(50*50),ncol=50)
for (i in 1:50){ for (j in 1:50) 
      {tden[i,j]=exp(-0.5*(A*(x1[i]^2)*(x2[j]^2)+(x1[i]^2)+(x2[j]^2)-2*B*x1[i]*x2[j]-2*C1*x1[i]-2*C2*x2[j]))}
}
tden=tden/sum(tden)
image(x1,x2,tden,col=terrain.colors(100),xlab="",ylab="")
contour(x1,x2,tden,add=TRUE,drawlabels=FALSE)
title("True Density")
}

[Package bayesm version 2.2-2 Index]