View of an Exchangeable Prior

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).

pgexchprior=function(lambda,pars)
{
alpha=pars[1]
a=pars[2]
b=pars[3]
(alpha-1)*log(prod(lambda))-(2*alpha+a)*log(alpha*sum(lambda)+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.)

alpha=c(5,20,80,400)
par(mfrow=c(2,2))
for (j in 1:4)
{
mycontour(pgexchprior,c(.001,5,.001,5),c(alpha[j],10,10))
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.