Quickest Search Over Multiple Sequences

• Published in: IEEE transactions on information theory Vol. 57; no. 8; pp. 5375 - 5386
• Main Authors: Lifeng Lai, Poor, H. V., Yan Xin, Georgiadis, G.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Bayesian; Bayesian analysis; CUSUM; Delay; Error probability; Exact sciences and technology; Formulations; Information theory; Information, signal and communications theory; Markov processes; optimal stopping; Optimization; Plugs; Probability distribution; quickest search; Random variables; Samples; Search problems; Searching; sequential analysis; Statistical analysis; Switches; Telecommunications and information theory; Testing; Tradeoffs; Performance evaluation; optimal stopping; CUSUM; Probabilistic approach; Change detection; sequential analysis; Bayesian; Probability distribution; Change point; Decision rule; False alarm rate; quickest search; Optimal decision; Optimal solution; Random variable
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2011.2159038

The problem of sequentially finding an independent and identically distributed sequence that is drawn from a probability distribution Q 1 by searching over multiple sequences, some of which are drawn from Q 1 and the others of which are drawn from a different distribution Q 0 , is considered. In the problem considered, the number of sequences with distribution Q 1 is assumed to be a random variable whose value is unknown. Within a Bayesian formulation, a sequential decision rule is derived that optimizes a trade-off between the probability of false alarm and the number of samples needed for the decision. In the case in which one can observe one sequence at a time, it is shown that the cumulative sum (CUSUM) test, which is well-known to be optimal for a non-Bayesian statistical change-point detection formulation, is optimal for the problem under study. Specifically, the CUSUM test is run on the first sequence. If a reset event occurs in the CUSUM test, then the sequence under examination is abandoned and the rule switches to the next sequence. If the CUSUM test stops, then the rule declares that the sequence under examination when the test stops is generated by Q 1 . The result is derived by assuming that there are infinitely many sequences so that a sequence that has been examined once is not retested. If there are finitely many sequences, the result is also valid under a memorylessness condition. Expressions for the performance of the optimal sequential decision rule are also developed. The general case in which multiple sequences can be examined simultaneously is considered. The optimal solution for this general scenario is derived.