In this paper, we present a Bayesian method for statistical model checking (SMC) of probabilistic hyperproperties specified in the logic HyperPCTL* on discrete-time Markov chains (DTMCs). While SMC of HyperPCTL* using sequential probability ratio test (SPRT) has been explored before, we develop an alternative SMC algorithm based on Bayesian hypothesis testing. In comparison to PCTL*, verifying HyperPCTL* formulae is complex owing to their simultaneous interpretation on multiple paths of the DTMC. In addition, extending the bottom-up model-checking algorithm of the non-probabilistic setting is not straight forward due to the fact that SMC does not return exact answers to the satisfiability problems of subformulae, instead, it only returns correct answers with high-confidence. We propose a recursive algorithm for SMC of HyperPCTL* based on a modified Bayes' test that factors in the uncertainty in the recursive satisfiability results. We have implemented our algorithm in a Python toolbox, HyProVer, and compared our approach with the SPRT based SMC. Our experimental evaluation demonstrates that our Bayesian SMC algorithm performs better both in terms of the verification time and the number of samples required to deduce satisfiability of a given HyperPCTL* formula.

本文介绍了基于贝叶斯方法的统计模型检查（SMC）算法，用于在离散时间马尔可夫链（DTMC）上检查指定于逻辑HyperPCTL*的概率超性质；我们提出了一种基于递归算法的SMC超算法，该算法基于修改后的贝叶斯测试，考虑了递归可满足结果的不确定性，并在一个Python工具箱HyProVer中实现；我们的实验评估表明，与基于SPRT的SMC相比，我们的贝叶斯SMC算法在验证时间和推断给定HyperPCTL*公式的可满足条件所需的样本数量方面表现更好。

使用HyperPCTL*的多智能体系统贝叶斯统计模型检验