Wednesday, August 20, 2014

Converting combination of random variables to hierarchical form for JAGS (BUGS, Stan, etc.)

An emailer asks:

Hi, John. Long-time listener, first-time caller... I have a model that says X is a function of three (independent) random variables:
X ~ normal(mu, sigma) / uniform(a,a+b) - beta(v,w)
and I also have N random samples of X. Can I use JAGS to estimate mu, sigma, a, b, v and w? Or is this problem outside its scope? Not sure if it's even possible.

My reply:

In other words, you want

x <- r / u - b
r ~ normal( rMean , rSD )
u ~ uniform( uLow , uHigh )
b ~ beta( bA , bB )

but the problem is that JAGS won't accept that sort of model specification.

One solution is to change to an equivalent hierarchical formalization:

x ~ normal( rMean/u-b , rSD/u )
u ~ uniform( uLow , uHigh )
b ~ beta( bA , bB )

This hierarchical form is directly implementable in JAGS. But be careful that in JAGS the normal is parameterized by precision, not SD, so you'll need to write x ~ dnorm( rMean/u-b , 1/(rSD/u)^2 ).

To demonstrate that the two forms are equivalent, here is a Monte Carlo sample generated in R from each formalization:

# Specify parameter values and sample size:
rMean=7 ; rSD=2
uLow=3 ; uHigh=5
bA=4 ; bB=3
N = 50000

# Sample x values your original way:
b = rbeta(N, bA,bB)
u = runif(N, uLow,uHigh)
r = rnorm(N, rMean,rSD)
x1 = r/u-b

# Sample x values a new way:
x2 = rep(0,N)
for ( i in 1:N ) {
  b = rbeta(1, bA,bB)
  u = runif(1, uLow,uHigh)
  x2[i] = rnorm(1, rMean/u-b , rSD/u )

# Plot results:
text( mean(x1) , 0 , "+" , cex=2 )
text( mean(x2) , 0 , "+" , cex=2 )

The result is shown below, where you can see that the original and new distributions are identical:

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