# A hierarchical Bayesian model using multinomial and Dirichlet distributions in JAGS

I am currently trying to model the state of a genetic locus in bacteria (which may be one of six values) using a hierarchical Bayesian model. This allows me to account for the fact that within a sample there is heterogeneity, as well as there being heterogeneity within a tissue type.

This is also good because:

- I am able to incorporate results from existing papers as priors.
- From the output I can quantify all the uncertainties within samples and tissues, check for significantly different distributions between condition types.

I think what I have ended up working with is probably something that could also be called a mixture model.

At any rate, I wrote some R code (available on my github) that uses rjags to interface with JAGS. I started off with code based on exercise 9.2 of ‘Doing Bayesian Data Analysis’ (Kruschke 2011, 1st ed.), which models the bias of multiple coins from multiple mints using data from the number of heads observed from each coin.

I then extended this to more than two outcomes (success, fail), which use a binomial distribution for the data and a beta distribution for the priors, to six outcomes. I now use a multinomial distribution for the data and Dirichlet distributions for the priors.

However I ran into an array of error messages when implementing this:

```
RUNTIME ERROR:
Compilation error on line 10.
Dimension mismatch taking subset of pi
RUNTIME ERROR:
Compilation error on line 16.
Dimension mismatch taking subset of alpha
RUNTIME ERROR:
Dimension mismatch when setting bounds in distribution ddirch
```

The first two of these was solved by getting all the brackets correct and consistent when tranisitoning from scalars -> vectors and vectors -> matrices. However the last message, ‘Dimension mismatch when setting bounds in distribution ddirch’, proved more insidious.

As JAGS is pretty closely based on BUGS, I searched for the problem in that domain (which seems to be a big advantage of JAGS compared to e.g. STAN – BUGS has been around for so long most problems have already arisen and been solved) and came across the following stackexchange/overflow issues that allowed me to solve the problem: http://stackoverflow.com/questions/24349692/dirichlet-multinomial-winbugs-code http://stats.stackexchange.com/questions/76644/define-priors-for-dirichlet-distribution-parameters-in-jags

The problem is that JAGS doesn’t allow the parameters of ddirch to be inferred. However you can use the observation that if $$delta[k] ~ dgamma(alpha[k], 1), then the vector with elements delta[k] / sum(delta[1:K]), k = 1, …, K, is Dirichlet with parameters alpha[k], k = 1, …, K.’ Then infer in the gamma distribution instead!

While you can find the full code here, the final working JAGS model code is pasted below. Note the use of the above trick, while using nested indices to allow for the hierarchical structure.

```
# JAGS model specification
model {
# For each sample, likelihood and prior
for (sample_index in 1:num_samples)
{
# likelihood
# Number of reads that map to each allele is multinomial.
# y and pi are matrices with num_samples rows and num_alleles columns
y[sample_index,1:num_alleles] ~ dmulti(pi[sample_index,1:num_alleles], N[sample_index])
# Dirichlet prior for proportion of population with allele in each sample
#
# This would be written like this:
# pi[sample_index,1:num_alleles] ~ ddirch(alpha[tissue[sample_index],1:num_alleles])T(0.0001,0.9999)
# Except JAGS doesn't allow the parameters of ddirch to be inferred
#
# Instead use the observation in the BUGS manual, and infer from a dgamma instead:
# 'The trick is to note that if delta[k] ~ dgamma(alpha[k], 1), then the vector with elements
# delta[k] / sum(delta[1:K]), k = 1, ..., K, is Dirichlet with parameters alpha[k], k = 1, ..., K.'
for (allele_index in 1:num_alleles)
{
pi[sample_index, allele_index]
delta[sample_index, allele_index] ~ dgamma(alpha[tissue[allele_index],allele_index], 1)
}
}
# For each tissue (blood or csf) hyperpriors
for (tissue_index in 1:num_tissues)
{
# Convert a and b in beta prior, to mu and kappa
# mu = mean allele for tissue,
# kappa = how closely does sequence represent tissue - constant across all tissue types
for (allele_index in 1:num_alleles)
{
alpha[tissue_index,allele_index]
}
# hyperpriors for mu (beta dist)
mu[tissue_index,1:num_alleles] ~ ddirch(AlphaMu[])
}
# Kappa hyperprior (gamma dist - shape and rate)
kappa ~ dgamma(Skappa, Rkappa)
# Top level constants for hyperpriors
# This is a vector of length num_alleles
AlphaMu <- alpha_priors
# Gamma dist for kappa. First convert mean and sd to shape and rate
Skappa <- pow(meanGamma,2)/pow(sdGamma,2)
Rkappa <- meanGamma/pow(sdGamma,2)
meanGamma <- 10
sdGamma <- 10
}
```