Empirical Bayesian estimators for a Poisson process propagated in time

Sometimes one may observe a time series of counts of which the underlying process generating the counts is unobserved, but of interest, both for its own sake and as a mean to predict future observations. This paper describes how from observations of counts in time modelled as a Poisson process, the parameters of the underlying process, modelled as a normally distributed random walk, can be estimated in the empirical Bayesian framework by means of an EM algorithm using Akaike's Bayesian Information Criterion (ABIC) for assessing the goodness of fit. Estimates for the posterior;mean and variance for predictions are given and inference discussed. Extensions are discussed in the light of composite link models introduced by Thompson and Baker. The method has been applied in the 'back calculation' of AIDS cases in Europe, results of which are reported elsewhere (ANONYMOUS, 1994; DOWNS et al., 1997). From the latter, the predictions for France are reported in somewhat more detail, together with the most recent surveillance data.