Fitting mixture of distributions (Gaussians + Uniform) in Winbugs

Question

I'm trying to fit a mixture distribution model to a vector of values, the mixture needs to consist of 2 gaussians distribution and 1 uniform distribution. I am trying to implement this in Winbugs. I found plenty of example that used mixture of gaussians, but can't figure how to add the uniform. The code paster below is currently parametrize to fit a vectors of values scaled between zero and one, but I get "multiple definitions of node NSD[1]", so it seems my structure is still wrong. Any suggestions?

model{

   ## priors
    xmin~dunif(0,1)
    eps2~dunif(0,1)
    xmax<-min(xmin+eps2, 1)
    mu1~dunif(0,1)
    eps1~dunif(0,1)
    mu2<-min(mu1+eps1,1)

   sigma1 ~ dunif(0,.5)     
   sigma2 ~ dunif(0,.5)     
   tau1<-pow(sigma1,-2)
   tau2<-pow(sigma2,-2)
   alpha[1]<-1
   alpha[2]<-1
   alpha[3]<-1
   p.state[1:3]~ddirch(alpha[])

   for (t in 1:npts) {
     idx[t] ~ dcat(p.state[])   ##  idx is the latent variable and the parameter index
     x[t,1]~dnorm(mu1,tau1)
     x[t,2]~dnorm(mu2,tau2)
     x[t,3]~dunif(xmin,xmax) 

      NSD[t] <-x[t,idx[t]]    
      }
} 

Answer 1

You could try using an uninformative dnorm prior in place of the dunif prior, so that you can model the prior for NSD as ~ dnorm(mu[idx[t]], tau[idx[t]]). You'd need to truncate, though, so could set very low/high bounds for truncation when you want normal priors.

Maybe something like this:

model  {
  mu[1] ~ dunif(0, 1)
  mu[2] <- min(mu[1] + eps[1], 1)
  mu[3] <- 0.5
  eps[1] ~ dunif(0, 1)
  eps[2] ~ dunif(0, 1)
  sigma[1] ~ dunif(0,.5)     
  sigma[2] ~ dunif(0,.5)     
  tau[1] <- pow(sigma[1],-2)
  tau[2] <- pow(sigma[2],-2)
  tau[3] <- 0.000001
  left[1] <- -100 # something relatively very low
  left[2] <- -100 # something relatively very low
  left[3] ~ dunif(0, 1)
  right[1] <- 100 # something relatively very high
  right[2] <- 100 # something relatively very high
  right[3] <- min(left[3] + eps[2], 1)
  alpha[1] <- 1
  alpha[2] <- 1
  alpha[3] <- 1
  p.state[1:3] ~ ddirch(alpha[])

  for (t in 1:npts) {
    idx[t] ~ dcat(p.state[])
    NSD[t] ~ dnorm(mu[idx[t]], tau[idx[t]])T(left[idx[t]], right[idx[t]])  
  }
}

A truncated vague normal distribution should be roughly equivalent to a uniform distribution. We can compare the kernel densities of samples from a dnorm(0, 0.000001)T(0, 1) and a dunif(0, 1). Here I use JAGS from R, but the outcome for WinBUGS should be similar:

library(R2jags)
M <- '
model {
  y_tnorm ~ dnorm(0, 0.000001)T(0, 1)
  y_unif ~ dunif(0, 1)
}
'
out <- jags(list(), NULL, c('y_tnorm', 'y_unif'), textConnection(M), 1, 100000, 
            n.burnin=0, n.thin=1, DIC=FALSE)

plot(density(out$BUGSoutput$sims.matrix[, 'y_tnorm'], bw=0.001), main='')
lines(density(out$BUGSoutput$sims.matrix[, 'y_unif'], bw=0.001), col=2)
legend('bottomright', c('Truncated normal', 'Uniform'), bty='n', 
       col=1:2, lty=1, inset=0.05)

---

EDIT

The model seems to run fine in JAGS.

M <- 'model  {
  mu[1] ~ dunif(0, 1)
  mu[2] <- min(mu[1] + eps[1], 1)
  mu[3] <- 0.5
  eps[1] ~ dunif(0, 1)
  eps[2] ~ dunif(0, 1)
  sigma[1] ~ dunif(0,.5)     
  sigma[2] ~ dunif(0,.5)     
  tau[1] <- pow(sigma[1],-2)
  tau[2] <- pow(sigma[2],-2)
  tau[3] <- 0.000001
  left[1] <- -100 # something relatively very low
  left[2] <- -100 # something relatively very low
  left[3] ~ dunif(0, 1)
  right[1] <- 100 # something relatively very high
  right[2] <- 100 # something relatively very high
  right[3] <- min(left[3] + eps[2], 1)
  alpha[1] <- 1
  alpha[2] <- 1
  alpha[3] <- 1
  p.state[1:3] ~ ddirch(alpha[])

  for (t in 1:npts) {
    idx[t] ~ dcat(p.state[])
    NSD[t] ~ dnorm(mu[idx[t]], tau[idx[t]])T(left[idx[t]], right[idx[t]])  
  }
}'


d <- read.csv('NSD.csv')

library(R2jags)
jagsfit <- jags(list(NSD=d$NSD, npts=nrow(d)), NULL, 
                c('mu', 'sigma', 'eps', 'left', 'right', 'p.state'), 
                textConnection(M), 3, 50000)

I haven't let it run long enough for all parameters to fully converge, but here's a preliminary look at some of your parameters.

##                  mean      sd       2.5%        25%        50%        75%      97.5%   Rhat n.eff
## deviance   -2650.2912 16.7002 -2667.7334 -2663.5577 -2656.7462 -2639.8387 -2610.2082 1.0054   450
## eps[1]         0.9514  0.0021     0.9472     0.9500     0.9514     0.9528     0.9556 1.0018  2500
## eps[2]         0.9100  0.0523     0.8438     0.8590     0.9018     0.9569     0.9975 1.0087   260
## left[1]     -100.0000  0.0000  -100.0000  -100.0000  -100.0000  -100.0000  -100.0000 1.0000     1
## left[2]     -100.0000  0.0000  -100.0000  -100.0000  -100.0000  -100.0000  -100.0000 1.0000     1
## left[3]        0.0021  0.0013     0.0001     0.0011     0.0021     0.0032     0.0043 1.0011 14000
## mu[1]          0.0008  0.0001     0.0007     0.0008     0.0008     0.0008     0.0009 1.0011 22000
## mu[2]          0.9522  0.0021     0.9480     0.9508     0.9522     0.9536     0.9564 1.0017  2600
## mu[3]          0.5000  0.0000     0.5000     0.5000     0.5000     0.5000     0.5000 1.0000     1
## p.state[1]     0.4721  0.0259     0.4217     0.4546     0.4721     0.4898     0.5227 1.0010 60000
## p.state[2]     0.3712  0.0265     0.3193     0.3532     0.3711     0.3890     0.4234 1.0017  2900
## p.state[3]     0.1567  0.0207     0.1189     0.1423     0.1558     0.1700     0.1999 1.0019  2300
## right[1]     100.0000  0.0000   100.0000   100.0000   100.0000   100.0000   100.0000 1.0000     1
## right[2]     100.0000  0.0000   100.0000   100.0000   100.0000   100.0000   100.0000 1.0000     1
## right[3]       0.9121  0.0522     0.8465     0.8610     0.9038     0.9589     0.9997 1.0087   260
## sigma[1]       0.0007  0.0000     0.0006     0.0007     0.0007     0.0007     0.0008 1.0010 60000
## sigma[2]       0.0238  0.0016     0.0210     0.0227     0.0238     0.0248     0.0272 1.0016  3200