Bayesian analysis of change-points in poisson processes

Abstract

Change-point analysis deals with the situation where an abrupt change has possibly taken place in the underlying mechanism that generates random variables. In a parametric setting, this means a change in the parameters of the underlying distribution. The interest is in whether such a change has actually taken place, and if it has, at which point in time. Also, there may have been more than one change during the period of interest. Application of change-point analysis is wide, but is particularly relevant in finance, the environment and medicine. The violability of markets may change abruptly, the rate and intensity of natural phenomena may change, or the effect of treatments in clinical trails may be studied. The literature on change-point problems is, by now, enormous. In this study we consider only the so-called non-sequential or fixed sample size version, although an informal sequential procedure, which follows from Smith (1975), is a routine consequence. Still, literature is substantial and our focus is on a fully Bayesian parametric approach. Use of the Bayesian framework for inference with regard to the change-point dates to work by Chernoff and Zacks (1964). Smith (1975) presents the Bayesian formulation for a finite sequence of independent observations. See also Zacks (1983). In our study we will consider only Poisson sequences and will address four situations: 1) When it is assumed that there is exactly one change-point, and proper priors are used. This can be generalised to more than one change-point. If the number of change-points is fixed and known, improper priors are also valid as will be explained later. 2) When there is a fixed number of change-points, the Markov Chain Monte Carlo method of Chib (1998) is useful, especially for large samples and multiple changepoints This approach will be described and applied. 3) When the number of change-points is unknown, and we want posterior probability distributions of the number of change-points, only proper priors are valid for calculating Bayes factors. In the case when no prior information is available, improper priors will cause the Bayes factor to have an indeterminate constant. In this case we apply the Fractional Bayes factor method of O’Hagan (1995). 4) When the data consists of multiple sequences, it is called multi-path changepoint analysis, and the distribution from which the change-points are drawn is of interest. Here the posterior distributions of parameters are estimated by MCMC methods. All the techniques are illustrated using simulated and real data sets.