  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.
