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Quickest Search Over Multiple Sequences

机译：多个序列上最快的搜索

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摘要

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 scenari-n-no is derived.
机译：通过搜索多个序列来顺序查找从概率分布$ Q_ {1} $中提取的独立且分布均匀的序列的问题，其中一些序列是从$ Q_ {1} $中提取的，而其他序列是从$ Q_ {1} $中提取的考虑不同的分布$ Q_ {0} $。在所考虑的问题中，具有分布$ Q_ {1} $的序列数假定为其值未知的随机变量。在贝叶斯公式中，导出了一个顺序决策规则，该规则优化了虚警概率与决策所需样本数量之间的权衡。在可以一次观察一个序列的情况下，已表明众所周知的累积和（CUSUM）检验对于非贝叶斯统计变化点检测公式是最佳的，研究中的问题。具体来说，CUSUM测试在第一个序列上运行。如果在CUSUM测试中发生重置事件，那么将放弃正在检查的序列，规则将切换到下一个序列。如果CUSUM测试停止，则规则声明测试停止时正在检查的序列由$ Q_ {1} $生成。通过假设存在无限多个序列来得出结果，这样就不会对已检查一次的序列进行重新测试。如果序列有限，则结果在无记忆条件下也是有效的。还开发了用于执行最佳顺序决策规则的表达式。考虑了可以同时检查多个序列的一般情况。推导了该一般场景的最佳解决方案。

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