Optimal Regret Exponents for Bayesian Statistical Decision Problems

2026-06-07

AI Summary

This study investigates the asymptotic behavior of minimal regret in Bayesian statistical decision problems with finite state and action spaces. By introducing multivariate Chernoff information and analyzing incompatible subsets of states, the work establishes—for the first time—an exact expression for the exponential decay rate of Bayesian regret under arbitrary loss functions. This result not only unifies and generalizes the classical Chernoff exponent theory from multi-hypothesis testing but also yields precise regret exponents for complex settings such as list hypothesis testing. The analysis reveals the fundamental mechanism underlying the exponential decay of regret under optimal strategies, providing a comprehensive characterization of the asymptotic optimality in Bayesian sequential decision-making.

Abstract

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.

Problem

• Bayesian decision theory
• regret exponent
• Chernoff information
• hypothesis testing
• statistical decision problems

Innovation

• Bayes regret
• error exponent
• Chernoff information
• hypothesis testing
• statistical decision theory