Active sequential hypothesis testing

Consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration. Due to the sequential nature of the problem, the decision maker relies on his current information state to adaptively select the most ``informative'' sensing action among the available ones. In this paper, using results in dynamic programming, lower bounds for the optimal total cost are established. The lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability. Moreover, upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions. It is shown that the first proposed heuristic achieves asymptotic optimality, where the notion of asymptotic optimality, due to Chernoff, implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration (hence the number of collected samples) increases. The second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search. However, by considering the dependency on the number of hypotheses, under a technical condition, this second heuristic is shown to achieve a nonzero information acquisition rate, establishing a lower bound for the maximum achievable rate and error exponent. In the case of a noisy dynamic search with size-independent noise, the obtained nonzero rate and error exponent are shown to be maximum.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1144 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Active Anomaly Detection in Heterogeneous Processes

An active inference problem of detecting anomalies among heterogeneous processes is considered. At each time, a subset of processes can be probed. The objective is to design a sequential probing strategy that dynamically determines which processes to observe at each time and when to terminate the search so that the expected detection time is minimized under a constraint on the probability of misclassifying any process. This problem falls into the general setting of sequential design of experiments pioneered by Chernoff in 1959, in which a randomized strategy, referred to as the Chernoff test, was proposed and shown to be asymptotically optimal as the error probability approaches zero. For the problem considered in this paper, a low-complexity deterministic test is shown to enjoy the same asymptotic optimality while offering significantly better performance in the finite regime and faster convergence to the optimal rate function, especially when the number of processes is large. The computational complexity of the proposed test is also of a significantly lower order.Comment: This work has been accepted for publication on IEEE Transactions on Information Theor

Quickest anomaly detection: A case of active hypothesis testing

Abstract — The problem of quickest detection of an anomalous process among M processes is considered. At each time, a subset of the processes can be observed, and the observations follow two different distributions, depending on whether the process is normal or abnormal. The objective is a sequential search strategy that minimizes the expected detection time subject to an error probability constraint. This problem can be considered as a special case of active hypothesis testing first considered by Chernoff in 1959, where a randomized test was proposed and shown to be asymptotically optimal. For the special case considered in this paper, we show that a simple deterministic test achieves asymptotic optimality and offers better performance in the finite regime. Index Terms—Sequential detection, hypothesis testing, dy-namic search. I

Asymptotically Optimal Anomaly Detection via Sequential Testing

Sequential detection of independent anomalous processes among K processes is considered. At each time, only M processes can be observed, and the observations from each chosen process follow two different distributions, depending on whether the process is normal or abnormal. Each anomalous process incurs a cost per unit time until its anomaly is identified and fixed. Switching across processes and state declarations are allowed at all times, while decisions are based on all past observations and actions. The objective is a sequential search strategy that minimizes the total expected cost incurred by all the processes during the detection process under reliability constraints. Low-complexity algorithms are established to achieve asymptotically optimal performance as the error constraints approach zero. Simulation results demonstrate strong performance in the finite regime.Comment: 28 pages, 5 figures, part of this work will be presented at the 52nd Annual Allerton Conference on Communication, Control, and Computing, 201