Quickest Search over Multiple Sequences with Mixed Observation

The problem of sequentially finding an independent and identically distributed (i.i.d.) sequence that is drawn from a probability distribution $f_1$ by searching over multiple sequences, some of which are drawn from $f_1$ and the others of which are drawn from a different distribution $f_0$, is considered. The observer is allowed to take one observation at a time. It has been shown in a recent work that if each observation comes from one sequence, the cumulative sum test is optimal. In this paper, we propose a new approach in which each observation can be a linear combination of samples from multiple sequences. The test has two stages. In the first stage, namely scanning stage, one takes a linear combination of a pair of sequences with the hope of scanning through sequences that are unlikely to be generated from $f_1$ and quickly identifying a pair of sequences such that at least one of them is highly likely to be generated by $f_1$. In the second stage, namely refinement stage, one examines the pair identified from the first stage more closely and picks one sequence to be the final sequence. The problem under this setup belongs to a class of multiple stopping time problems. In particular, it is an ordered two concatenated Markov stopping time problem. We obtain the optimal solution using the tools from the multiple stopping time theory. The optimal solution has a rather complex structure. For implementation purpose, a low complexity algorithm is proposed, in which the observer adopts the cumulative sum test in the scanning stage and adopts the sequential probability ratio test in the refinement stage. The performance of this low complexity algorithm is analyzed when the prior probability of $f_{1}$ occurring is small. Both analytical and numerical simulation results show that this search strategy can significantly reduce the searching time when $f_{1}$ is rare.