A simple Bayesian approach to multiple change-points

Abstract

After a brief review of previous frequentist and Bayesian approaches to multiple change-points, we describe a Bayesian model for multiple parameter changes in a multiparameter exponential family. This model has attractive statistical and computational properties and yields explicit recursive formulas for the Bayes estimates of the piecewise constant parameters. Efficient estimators of the hyperparameters of the Bayesian model for the parameter jumps can be used in conjunction, yielding empirical Bayes estimates. The empirical Bayes approach is also applied to solve long-standing frequentist problems such as significance testing of the null hypothesis of no change-points versus multiple change-point alternatives, and inference on the number and locations of change-points that partition the unknown parameter sequence into segments of equal values. Simulation studies of performance and an illustrative application to the British coal mine data are also given. Extensions from the exponential family to general parametric families and from independent observations to genearlized linear time series models are then provided.

A Bayesian Approach to Sequential Surveillance in Exponential Families

“…2. As noted in Lai and Xing (2008b), one can apply the same change-point model and use the same ideas to develop recursive estimators for more general parametric families f y t than the exponential family (12). In particular, corresponding to a prior density function that is proportional to g , the conditional density function g i t of t given K t = i and t is g i t ∝ g t k=i f y i and therefore (19) can be generalized to f t t = t i=1 p it g i t t , with p it = p * it / t k=i p * kt , where p * it is given by (20) but with 1/ 0 0 = g d 1/ ij = j k=i f y k g d…”

“…Each entry in the table is based on 1,000 simulations; the standard errors are shown in parentheses. Closely related to these results is an asymptotic optimality theory for BCMIX-APE in estimating t ; see Lai and Xing (2008b) for details. We also apply to these simulated samples the detection rule (23), in which p = 0 001, k p = 30 and the threshold is chosen to give a false alarm rate of approximately 0.001; see Lai (1995, p. 628) and Lai and Xing (2008c) for details concerning how can be determined by Monte Carlo methods to satisfy a prescribed false alarm rate.…”

“…3 on implementation issues. For small p, the following approximation called BCMIX (bounded complexity mixture) has been introduced by Lai and Xing (2008b) to the recursive formulas for density (19) can be obtained as follows. Let t−1 p be the set of indices i for which p i t−1 is kept at stage t − 1; thus, t−1 p ⊃ t − 1 t − m p .…”

Sequential Change-Point Detection When the Pre- and Post-Change Parameters are Unknown

“…From (3.17), it follows that (2.7) with suitably chosen k p is asymptotically equivalent to the window-limited GLR rule. This asymptotic equivalence actually holds more generally for the prior densities assumed in the theorem because Laplace's asymptotic formula for integrals can be used for more general prior densities of 0 1 than conjugate priors that are independent, as pointed out in Section 6 of Lai and Xing (2008).…”

A Novel Bayesian Change-point Algorithm for Genome-wide Analysis of Diverse ChIPseq Data Types

“…To meet these objectives, we first constructed a statistical framework that naturally modeled ChIPseq data structures using a cutting edge advance in HMMs 9 , which utilizes only explicit formulas-an innovation crucial to its performance advantages. More sophisticated then heuristic models, our HMM accommodates infinite hidden states through a Bayesian model.…”