### 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.