SUMMARY A general approach to hierarchical Bayes changepoint models is presented. In particular, desired marginal posterior densities are obtained utilizing the Gibbs sampler, an iterative Monte Carlo method. This approach avoids sophisticated analytic and numerical high dimensional integration procedures. We include an application to changing regressions, changing Poisson processes and changing Markov chains. Within these contexts we handle several previously inaccessible problems.

Hierarchical Bayesian Analysis of Changepoint Problems

In adult life, people normally know what they are doing. This experience of controlling one's own actions and, through them, the course of events in the outside world is called 'sense of agency'. It forms a central feature of human experience; however, the brain mechanisms that produce the sense of agency have only recently begun to be investigated systematically. This recent progress has been driven by the development of better measures of the experience of agency, improved design of cognitive and behavioural experiments, and a growing understanding of the brain circuits that generate this distinctive but elusive experience. The sense of agency is a mental and neural state of cardinal importance in human civilization, because it is frequently altered in psychopathology and because it underpins the concept of responsibility in human societies.

Sense of agency in the human brain

Inference about the Change-Point in a Sequence of Random Variables: A Selection Approach

Changepoint identification is important in many data analysis problems, such as industrial control and medical diagnosis-given a data sequence, we wish to make inference about the location of one or more points of the sequence at which there is a change in the model or parameters driving the system. For long data sequences, however, analysis (especially in the multiple-changepoint case) can become computationally prohibitive, and for complex non-linear models analytical and conventional numerical techniques are infeasible. We discuss the use of a sampling-based technique, the Gibbs sampler, in multiple-changepoint problems and demonstrate how it can be used to reduce the computational load involved considerably

Bayesian Retrospective Multiple- changepoint Identification

Publisher Summary This chapter discusses fixed sample and the sequential procedures of testing and estimation and describes classical and Bayesian approaches to the change-point problem. It presents Bayesian and maximum likelihood estimation of the location of the shift points. The Bayesian approach is based on modeling the prior distribution of the unknown parameters, adopting a loss function and deriving the estimator, which minimizes the posterior risk. The chapter discusses this approach with an example of a shift in the mean of a normal sequence. The estimators obtained are generally nonlinear complicated functions of the random variables. From the Bayesian point of view, these estimators are optimal. The maximum likelihood estimation of the location parameter of the change point is an attractive alternative to the Bayes estimators. Regression relationship can change at unknown epochs, resulting in different regression regimes that should be detected and identified.

Survey of classical and bayesian approaches to the change-point problem: fixed sample and sequential procedures of testing and estimation

The problem of estimating the switch point in a sequence of independent random variables is studied from a Bayesian viewpoint. Theoretical results and numerical examples are given for the normal sequence and two-phase regression.

Bayesian inferences related to shifting sequences and two-phase regression

This study is designed to investigate Bayesian procedures for estimating the time point at which a parameter change occurred in an observed sequence of independent random variables of the regular exponential class. In particular, binomial, exponential, and normal sequences are considered and a generalization to the so-called two-phase regression problem is emphasized. In addition, inference to other parameters of the sequence is made.

Bayesian inferences about a changing sequence of random variables

This study is a Bayesian analysis of a regression model with autocorrelated errors which exhibits one change in the regression parameters and where the autocorrelation parameter is unknown Using a normal-gamma prior for all the parameters except the shift point which has a uniform distribution, the marginal posterior distribution of the regression parameters, the shift point and the precision of the errors is found. It is important to know where the shift occurred thus the main emphasis is with the posterior distribution of the shift point A numerical study assesses the effect of the values of the shift point and the magnitude of the shift on the posterior distribution of the shift point. The posterior distribution of the shift point is more sensitive to change, which occurs in the middle of the observations than to one which occurs at an extreme of the data.

Parameter changes in a regression model with autocorrelated errors

The general mixed linear model, containing both the fixed and random effects, is considered. Using gamma priors for the variance components, the conditional posterior distributions of the fixed effects and the variance components, conditional on the random effects, are obtained. Using the normal approximation for the multiple t distribution, approximations are obtained for the posterior distributions of the variance components in infinite series form. The same approximation Is used to obtain closed expressions for the moments of the variance components. An example is considered to illustrate the procedure and a numerical study examines the closeness of the approximations.

Bayesian inference for the variance components in general mixed linear models

An essential ingredient of any time series analysis is the estimation of the model parameters and the forecasting of future observations. This investigation takes a Bayesian approach to the analysis of time series by making inferences of the model parameters from the posterior distribution and forecasting from the predictive distribution. The foundation of the approach is to approximate the condi-tional likelihood by a normal-gamma distribution on the parameter space. The techniques illustrated with many examples of ARMA processes.

Lyle D. Broemeling

Papers

Bayesian inferences related to shifting sequences and two-phase regression

Bayesian inferences about a changing sequence of random variables

Parameter changes in a regression model with autocorrelated errors

Bayesian inference for the variance components in general mixed linear models

Inference and prediction with arma processes