Aspects of Bayesian change-point analysis

Abstract

English: In chapter one we looked at the nature of structural change and defined structural change as a change in one or more parameters of the model in question. Bayesian procedures can be applied to solve inferential problems of structural change. Among the various methodological approaches within Bayesian inference, emphasis is put on the analysis of the posterior distribution itself, since the posterior distribution can be used for conducting hypothesis testing as well as obtaining a point estimate. The history of structural change in statistics, beginning in the early 1950's, is also discussed. Furthermore the Bayesian approach to hypothesis testing was developed by Jeffreys (1935, 1961), where the centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is one-half. According to Kass and Raftery (1993) this posterior odds = Bayes factor x prior odds and the Bayes factor is the ratio of the posterior odds of Hl to its prior odds, regardless of the value of the prior odds. The intrinsic and fractional Bayes factors are defined and some advantages and disadvantages of the IBF's are discussed. In chapter two changes in the multivariate normal model are considered. Assuming that a change has taken place, one will want to be able to detect the change and to estimate its position as well as the other parameters of the model. To do a Bayesian analysis, prior densities should be chosen. Firstly the hyperparameters are assumed known, but as this is not. usually true, vague improper priors are used (while the number of change-point.s is fixed). Another way of dealing with the problem of unknown hyperparameters is to use a hierarchical model where the second stage priors are vague. We also considered Gibbs sampling and gave the full conditional distributions for all the cases. The three cases that are studied is (1) a change in the mean with known or unknown variance, (2) a change in the mean and variance by firstly using independent prior densities on the different variances and secondly assuming the variances to be proportional and (3) a change in the variance. The same models above are also considered when the number of change-points are unknown. In this case vague priors are not appropriate when comparing models of different dimensions. In this case we revert to partial Bayes factors, specifically the intrinsic and fractional Bayes factors, to obtain the posterior probabilities of the number of change-points. Furthermore we look at component analysis, i.e. determining which components of a multivariate variable are mostly responsible for the changes in the parameters. The univariate case is then also considered in more detail, including multiple model comparisons and models with auto correlated errors. A summary of approaches in the literature as well as four examples are included. In chapter three changes in the linear model, with (1) a change in the regression coefficient and a constant variance, (2) a change in only the variance and (3) a change in the regression coefficient and the variance, are considered. Bayes factors for the above mentioned cases, multiple change-points, component analysis, switchpoint (continuous change-point) and auto correlation are included, together with seven examples. In chapter four changes in some other standard models are considered. Bernoulli type experiments include the Binomial model, the Negative binomial model, the Multinomial model and the Markov chain model. Exponential type models include the Poisson model, the Gamma model and the Exponential model. Special cases of the Exponential model include the left truncated exponential model and the Exponential model with epidemic change. In all cases the partial Bayes factor is used to obtain posterior probabilities when the number of change-points is unknown. Marginal posterior densities of all parameters under the change-point model are derived. Eleven examples are included. In chapter five change-points in the hazard rate are studied. This includes an abrupt change in a constant hazard rate as well as a change from a decreasing hazard rate to a constant hazard rate or a change from a constant hazard rate to an increasing hazard rate. These hazard rates are obtained from combinations of Exponential and Weibull density functions. In the same way a bathtub hazard rate can also be constructed. Two illustrations are given. Some concluding remarks are made in chapter six, with discussions of other approaches in the literature and other possible applications not dealt with in this study.