Aspects of Bayesian changepoint 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 onehalf. 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 changepoint.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 changepoints 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 changepoints. 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 changepoints, component analysis, switchpoint
(continuous changepoint) 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 changepoints is unknown. Marginal posterior densities of all parameters
under the changepoint model are derived. Eleven examples are included.
In chapter five changepoints 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.
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