Although I may ask a question that is already solved (but I found none that would explicitly refer to this), I would like to know, if (and I am no statistician) I am right with programming the log-likelihood function of a beta-binomial distribution in R as follows:
sum(lgamma(theta) - lgamma(Pi * theta) - lgamma((1 - Pi) * theta) + lgamma(n + 1) - lgamma(Y + 1) - lgamma((n - Y) + 1) + lgamma(Y + Pi * theta) + lgamma(n - Y + (1 - Pi) * theta) - lgamma(n + theta))
Here, theta is used as an overdispersion parameter. n is the number of trials, Y are the actual successes (from the real data), and Pi is the probability of success as derived from my model in JAGS. I (naively) took the density function as stated in Bolker (2008) and tried to take the logarithm of it. After that, I used this to compute an AIC and it gave reasonable numbers, but I am not sure!
P.S. I know there are caveats about using AIC, especially in a bayesian context, but I just want to know if I am right in the way to derive the answer to this, although trying it may be foolish from a philosophical point of view in the first place...