Carlin-Gelfand-Smith (1992) Hierarchical Bayesian Analysis of Changepoint Problems

Summary

Carlin, Gelfand and Smith present a general approach to hierarchical Bayes changepoint models in which the desired marginal posterior densities are obtained by the Gibbs sampler rather than by sophisticated analytic or high-dimensional numerical integration. Working in the non-sequential (fixed-sample-size) setting, they show the same iterative Monte Carlo machinery handles changing regressions, changing Poisson processes, and changing Markov chains — making several previously inaccessible changepoint problems routine.

Key Claims

• Gibbs sampling for changepoints. Casting the changepoint problem as a hierarchical Bayes model and sampling the full conditionals of the changepoint location(s) and the segment-specific parameters yields the marginal posteriors directly, avoiding the analytic/numerical integration that previously limited such models.
• One engine, many models. The approach applies uniformly to changing regressions (a break in regression coefficients), changing Poisson processes (a break in an event rate, e.g. the coal-mining-disasters series), and changing Markov chains (a break in transition probabilities).
• Hierarchical structure. Priors on the changepoint and on the segment parameters (with hyperpriors) let information be shared and uncertainty propagated coherently, with the changepoint's marginal posterior computed as a by-product of the sampler.
• Previously inaccessible problems become routine. The generic sampler dissolves model-specific derivations, so richer changepoint models can be fit with little extra effort.

Concepts Introduced or Extended

• Change-Point Analysis

Entities Mentioned

• Bradley P. Carlin
• Alan E. Gelfand
• Adrian F. M. Smith

Quotes

"A general approach to hierarchical Bayes changepoint models is presented. In particular, desired marginal posterior densities are obtained utilizing the Gibbs sampler … This approach avoids sophisticated analytic and numerical high dimensional integration procedures."

My Take

A canonical early demonstration of the Gibbs-sampling revolution kicked off by Gelfand-Smith (1990): changepoint problems were a natural showcase because their awkward, discontinuous likelihoods had long forced bespoke analytics, and the paper shows a single sampler dispatching regressions, Poisson processes, and Markov chains alike. It is the fixed-$k$ Bayesian counterpart on the change-point page to the later trans-dimensional reversible-jump treatment: here the number of changepoints is known (or compared across models) and the sampler just needs the full conditionals, whereas RJMCMC lets $k$ itself move. Its lasting influence is less any specific model than the template — write the hierarchical model, read off the conditionals, sample — that became the default way applied Bayesians handle structural breaks, and it sits naturally beside the same-era medical Gibbs-sampling and frailty applications.