Bayesian predictive inference for time series count data

Correlated count data arise often in practice, especially in repeated measures situations or instances in which observations are collected over time. In this paper, we consider a parametric model for a time series of counts by constructing a likelihood-based version of a model similar to that of Zeger (1988, Biometrika 75, 621-629). The model has the advantage of incorporating both overdispersion and autocorrelation. We consider a Bayesian approach and propose a class of informative prior distributions for the model parameters that are useful for prediction. The prior specification is motivated from the notion of the existence of data from similar previous studies, called historical data, which is then quantified into a prior distribution for the current study. We derive the Bayesian predictive distribution and use a Bayesian criterion, called the predictive L measure, for assessing the predictions for a given time series model. The distribution of the predictive L measure is also derived, which will enable us to compare the predictive ability for each model under consideration. Our methodology is motivated by a real data set involving yearly pollen counts, which is examined in some detail.