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Asymptotically optimal Bayesian sequential change detection and identification rules
Cataloged from PDF version of article.We study the joint problem of sequential change detection and multiple hypothesis
testing. Suppose that the common distribution of a sequence of i.i.d. random variables
changes suddenly at some unobservable time to one of finitely many distinct alternatives,
and one needs to both detect and identify the change at the earliest possible time. We propose
computationally efficient sequential decision rules that are asymptotically either Bayesoptimal
or optimal in a Bayesian fixed-error-probability formulation, as the unit detection
delay cost or the misdiagnosis and false alarm probabilities go to zero, respectively. Numerical
examples are provided to verify the asymptotic optimality and the speed of convergence
On the Wiener disorder problem
In the Wiener disorder problem, the drift of a Wiener process changes
suddenly at some unknown and unobservable disorder time. The objective is to
detect this change as quickly as possible after it happens. Earlier work on the
Bayesian formulation of this problem brings optimal (or asymptotically optimal)
detection rules assuming that the prior distribution of the change time is
given at time zero, and additional information is received by observing the
Wiener process only. Here, we consider a different information structure where
possible causes of this disorder are observed. More precisely, we assume that
we also observe an arrival/counting process representing external shocks. The
disorder happens because of these shocks, and the change time coincides with
one of the arrival times. Such a formulation arises, for example, from
detecting a change in financial data caused by major financial events, or
detecting damages in structures caused by earthquakes. In this paper, we
formulate the problem in a Bayesian framework assuming that those observable
shocks form a Poisson process. We present an optimal detection rule that
minimizes a linear Bayes risk, which includes the expected detection delay and
the probability of early false alarms. We also give the solution of the
``variational formulation'' where the objective is to minimize the detection
delay over all stopping rules for which the false alarm probability does not
exceed a given constant.Comment: Published in at http://dx.doi.org/10.1214/09-AAP655 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Multisource Bayesian sequential change detection
Suppose that local characteristics of several independent compound Poisson
and Wiener processes change suddenly and simultaneously at some unobservable
disorder time. The problem is to detect the disorder time as quickly as
possible after it happens and minimize the rate of false alarms at the same
time. These problems arise, for example, from managing product quality in
manufacturing systems and preventing the spread of infectious diseases. The
promptness and accuracy of detection rules improve greatly if multiple
independent information sources are available. Earlier work on sequential
change detection in continuous time does not provide optimal rules for
situations in which several marked count data and continuously changing signals
are simultaneously observable. In this paper, optimal Bayesian sequential
detection rules are developed for such problems when the marked count data is
in the form of independent compound Poisson processes, and the continuously
changing signals form a multi-dimensional Wiener process. An auxiliary optimal
stopping problem for a jump-diffusion process is solved by transforming it
first into a sequence of optimal stopping problems for a pure diffusion by
means of a jump operator. This method is new and can be very useful in other
applications as well, because it allows the use of the powerful optimal
stopping theory for diffusions.Comment: Published in at http://dx.doi.org/10.1214/07-AAP463 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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