We study finite-state finite-action Bayesian statistical decision problems. While exact error-exponent characterizations are known for several special cases, including hypothesis testing and hypothesis exclusion, the asymptotic behavior of the optimal Bayes regret is largely unknown for general decision problems. In this paper, we show that the optimal regret always decays exponentially fast and characterize its exact exponent for arbitrary loss functions. The exponent is given by the minimum multivariate Chernoff information over the minimal incompatible subsets of states, where an incompatible subset is a collection of states for which no single action is optimal for all states in the subset. Our result recovers the classical pairwise-minimum Chernoff exponent for symmetric multiple hypothesis testing and the multivariate Chernoff exponent for hypothesis exclusion, while also yielding, to the best of our knowledge, the first exact exponent characterization for list hypothesis testing.

翻译：我们研究有限状态-有限动作的贝叶斯统计决策问题。尽管包括假设检验与假设排除在内的若干特例已存在精确的误差指数刻画，但一般决策问题中最优贝叶斯遗憾的渐近行为尚不明确。本文证明最优遗憾始终以指数速度衰减，并针对任意损失函数刻画其精确指数。该指数由所有最小不相兼容状态子集上的多元Chernoff信息最小值给出，其中不相兼容子集指代不存在单一动作能对所有状态均为最优的状态集合。该结果既统一了对称多元假设检验中的经典成对最小Chernoff指数与假设排除中的多元Chernoff指数，又首次实现了列表假设检验精确指数的刻画（据我们所知）。