2-7 are the prior distributions set for r and 2 respectively. Since the gamma prior distribution is
the conjugate prior for the Poisson distribution (Gelman et al., 1995), it is a very convenient
distribution that can be used, especially in a Bayesian Markov Chain Monte Carlo simulation.
I used the Bayesian Markov Chain Monte Carlo simulation approach using the Gibbs
sampler (Gelman et al., 1995) to determine the posterior distribution of the parameters r, 2 and
N,. The Gibbs sampler is a particular Markov chain algorithm useful in such multidimensional
problems based on alternate conditional sampling. To use the Gibbs sampler, the conditional
distributions of each parameter have to be derived by treating the other parameters as known
(full conditionals). The unnormalized joint posterior density function is
P(2,r,{Ni} | w) oc [If (w. I T,,r,N,)g(N, I )]P(r)P(2)
i
The objective is to sample from the joint posterior density function repeatedly and the
Markov chain that develops represents the joint posterior distribution. However, since this is a
hierarchical model and all the probabilities are not independent, an alternative is to sequentially
sample from each full conditional derived for each parameter. This is the whole purpose of the
Gibbs sampler.
The full conditionals are derived as follows:
Full conditional for A:
rR -1 (2-8)
[1 .]~ gamma a+ZN,, +R
,=1 b --
Full conditional for r:
P(r) = 1, 0