Sequential Bayesian Analysis of Multivariate Poisson Count Data

In this paper, we develop a new class of dynamic multivariate time series models for count data and consider their sequential inference for online updating. In doing so, we provide and show how the class can be constructed from a conjugate static univariate case. The key property of the model is its ability to model serial dependence of counts via a state space structure while allowing for dependence across multiple series by assuming a common random environment. Other notable features include analytic forms for propagation and predictive likelihood densities. Sequential updating occurs via sufficient statistics for static model parameters, all of which lead to a fully adapted implementation of our algorithm. Furthermore, we show that the proposed model leads to a new class of predictive likelihoods (marginals) which we refer to as the (dynamic) multivariate confluent hyper-geometric negative binomial distribution (MCHG-NB) and a new multivariate distribution which we call the dynamic multivariate negative binomial (DMNB) distribution. To illustrate our methodology, we use two simulated data sets as well as a data set on weekly household consumer demand of consumer non-durable goods.