Steve Brooks writes,

Suppose you’re fitting a simple generalized linear model with two different fixed effects each with several different levels e.g.,

Y_{ijk} ~ Poisson (L_{ij})
and log L_{ij} = a + b_i + c_j

For identifiability you need to fix a couple of parameters (e.g., b_1 = c_1 =0), but the choice of which to fix and to what value is arbitrary and will affect all of the other parameter values. This then means that the penalty from the priors differs depending on what constraints you impose and though this may not have much affect on posterior means etc. it can make a huge difference to the associated posterior model probabilities. In particular, you can calculate the model probabilities between two identical models but with different constraints (thus identical likelihood, but effectively different priors) and you get non-equal PMP’s.

This is pretty basic and well-known, right, but is there a general consensus on what to do about it? If you want to compare models, there are two things you can do: (1) essentially average over all possible constraints, but then what’s the theoretical justification for this; and (2) find a prior that is invariant to the constraint, but this is often tricky.

My reply:

What I would do is to model the b’s and the c’s (for example, b_i ~ N(mu_b, sigma^2_b), and c_j ~ N(mu_c,sigma^2_c)). The model now has redundant parameters, so then I’d identify things by defining:

a.adj <- a + mean(b[]) + mean(c[]) b.adj[i] <- b[i] - mean(b[]), for each i c.adj[j] <- c[j] - mean(c[]), for each j (For convenience I'm using Bugs notation.) These b.adj's and c.adj's are what I call finite-population effects. We discuss it more in our forthcoming book. The above is the cleanest approach, I think.