A Bayesian approach to two-sided quickest change detection

The problem of detecting an abrupt change in a sequence of independent and identically distributed (IID) random variables is addressed. Sequentially received samples are IID both before and after a single unknown change time. Unlike previous approaches to change detection that assume a known probability density function (PDF) for the observations at the start, the problem formulated here is to detect a change between two given PDFs in either direction, meaning that at any given time the number of hypotheses to be tracked is always twice the number of samples received. A Bayesian multiple hypothesis approach is proposed and shown to have the following properties: (i) unlike previous tests that operate with a threshold, the minimum-cost hypothesis is tracked through time, including that of no change. (ii) under an exponential delay cost function and suitable parameter choices, the proposed procedure's probability of detecting a change in the incorrect direction asymptotically vanishes with time, (iii) the method is recursive with constant computation per unit time, and (iv) error probabilities may be directly traded off with average delay. Performance results using simulation confirm the derived properties and also reveal that the additional average delay after a transient period corresponding to when the starting state is uncertain, compared to that of the optimal one-sided test, CUSUM, is modest.