# Bayesian hypothesis testing for multinomial-Dirichlet

I am working on estimating parameters for a behavioural experiment where the likelihood of my data is modelled as a multinomial with the parameters summing to 1, and I have a weakly informative Dirichlet prior. I have done the usual posterior predictive checks and the multinomial-Dirichlet model seems to work much better than a Gaussian model with log-transformed data.

I am unsure what a good test statistic for hypothesis testing (comparing MCMC posterior distributions for "control" versus "mutant") would be in this case. I'd appreciate help on some sort of a statistic which I could use to arrive at something akin to a $p$-value. Thanks in advance.