In Chapter 7, we consider the following exchangeable prior for Poisson rates lam_1, ..., lam_k that is described in two stages.
Stage I. Conditional on parameters alpha, mu, lam_1, ..., lam_k are independent Gamma(alpha, alpha/mu)
Stage II. The parameters (alpha, mu) come from a specified prior g(alpha, mu).
Here mu is the prior mean of lam_i and alpha is a precision parameter. This structure induces the following prior on lam_1, .., lam_k:
g(lam_1, ..., lam_k) = integral prod P(lam_j | alpha, mu) g(alpha, mu) dalpha dmu.
To see how this prior reflects dependence between the parameters, suppose we fix alpha to the value alpha_0 and let mu be distributed inverse gamma(a, b). Then one can show the prior on lam_1,..., lam_k is given (up to a proportionality constant) by
g(lam_1, ..., lam_k) = P^(alpha_0-1)/(alpha_0 S + b)^(k alpha_0 + a),
where P is the product of lam_j and S is the sum of lam_j.
To see this prior, we program a simple function pgexchprior that computes the logarithm of the prior of lam_1 and lam_2 given parameter values (alpha_0, a, b).
The following R commands construct contour plots of the prior for lam_1 and lam_2 for the precision parameters alpha_0 = 5, 20, 80, and 200. (In each case, we assign mu an inverse-gamma (10, 10) prior.)
for (j in 1:4)
title(main=paste("ALPHA = ",alpha[j]),xlab="LAMBDA1",ylab="LAMBDA2")
These plots clearly show that, as alpha increases, the prior induces stronger correlation between the two Poisson rates.