A Bayesian Approach to Sequential Surveillance in Exponential Families

Abstract

We describe herein a Bayesian change-point model and the associated recursive formulas for the estimated time-varying parameters and the posterior probability that a change-point has occurred at a particular time. The proposed model is a variant of that of Chernoff and Zacks (1964) for the case of normal means with known common variance. It considers more generally the multiparameter exponential family and addresses the complex statistical issues due to multiple change-points and unknown pre-and post-change system parameters in sequential surveillance. A sequential detection rule based on the proposed model is also introduced and its false alarm rate and mean detection delay are studied in the multiple change-point setting.

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

“…On the other hand, the sliding window idea introduced in Lai (1995) can be used to modify Shiryaevtype rules, as we have done in Section 2, thereby providing an asymptotically optimal solution to the Bayes problem in Section 3.2. In Lai et al (2009) further extended the Bayesian model from a single change-time to a sequence of change-times 1 2 with independent and geometrically distributed increments that have a common parameter p thereby extending the sequential changepoint detection rule to a sequential surveillance rule. The scenario of unknown preand post-change parameters is of common occurence in surveillance applications, whereas the baseline (in-control) parameter value is usually specified in quality control or fault detection.…”

“…Lai et al (2009) have modified the above Bayesian model and detection rule to handle the case where multiple change-points with unknown pre-and post-change parameters can occur, as in sequential surveillance applications. The key idea is still to use sliding windows n i=n−k p as in (2.7) but with the summands modified to be the posterior probabilities p in that the most recent change-point up to time n occurs at i.…”

Sequential generalized likelihood ratio tests for vaccine safety evaluation

“…These methods include the sequential GLR detection rules, which have been extended in [31] to tackle the case where both the baseline and post-change parameters are unknown. Moreover, a theory of sequential surveillance, comparable to the relatively complete theory of sequential detection, is introduced in [30] for exponential families. While sequential testing can be used to test if the adverse event rate of a large cohort of vaccinated subjects in a post-licensure study differs from the baseline rate, sequential surveillance can be used to detect elevated risks due to environmental, viral or other changes that have affected the approved vaccine's safety and efficacy, or for certain sub-populations with previously undetected risk factors.…”

A simple Bayesian approach to multiple change-points

“…For on-line (sequential) estimation and detection problems, the BCMIX filter can be extended beyond the exponential family to provide recursive estimators and detection schemes. In connection with ongoing work in this direction, Lai, Liu, and Xing (2009) have made use of BCMIX filters to develop efficient sequential surveillance schemes for detecting multiple parameter changes in multivariate exponential families. In Section 6 we extended the change-point model and its associated methodology from the independent exponential family to generalized linear autoregressive models whose regression and dispersion parameters may undergo occasional jumps.…”