Posterior-Based Stopping Rules for Bayesian Ranking-and-Selection Procedures

Abstract

Sequential ranking-and-selection procedures deliver Bayesian guarantees by repeatedly computing a posterior quantity of interest—for example, the posterior probability of good selection or the posterior expected opportunity cost—and terminating when this quantity crosses some threshold. Computing these posterior quantities entails nontrivial numerical computation. Thus, rather than exactly check such posterior-based stopping rules, it is common practice to use cheaply computable bounds on the posterior quantity of interest, for example, those based on Bonferroni’s or Slepian’s inequalities. The result is a conservative procedure that samples more simulation replications than are necessary. We explore how the time spent simulating these additional replications might be better spent computing the posterior quantity of interest via numerical integration, with the potential for terminating the procedure sooner. To this end, we develop several methods for improving the computational efficiency of exactly checking the stopping rules. Simulation experiments demonstrate that the proposed methods can, in some instances, significantly reduce a procedure’s total sample size. We further show these savings can be attained with little added computational effort by making effective use of a Monte Carlo estimate of the posterior quantity of interest. Summary of Contribution: The widespread use of commercial simulation software in industry has made ranking-and-selection (R&S) algorithms an accessible simulation-optimization tool for operations research practitioners. This paper addresses computational aspects of R&S procedures delivering finite-time Bayesian statistical guarantees, primarily the decision of when to terminate sampling. Checking stopping rules entails computing or approximating posterior quantities of interest perceived as being computationally intensive to evaluate. The main results of this paper show that these quantities can be efficiently computed via numerical integration and can yield substantial savings in sampling relative to the prevailing approach of using conservative bounds. In addition to enhancing the performance of Bayesian R&S procedures, the results have the potential to advance other research in this space, including the development of more efficient allocation rules.