The Type Theory of Stationary MDPs: Rare Events and Uncertainty Quantification

The method of types, pioneered by Csiszar (1998), uses combinatorial arguments to approximate the probabilities of empirical measures and has become a standard tool for deriving error rates in hypothesis testing, source coding, and related information-theoretic problems. Although the technique is widely used across machine learning, computer science, and econometrics, rigorous type-theoretic guarantees for stationary Markov decision processes and reinforcement learning remain limited. This paper addresses that gap by developing a method-of-types framework for the empirical pair measure of the state--action process of a stationary Markov decision process. We establish finite-sample type bounds for the state--action process and characterize the associated entropy cost through the conditional Kullback--Leibler divergence relative to the true pair kernel. These results yield large deviation principles for learned MDP dynamics and variational rate functions for plug-in value- and Q-function estimators in offline reinforcement learning. We further develop finite-sample uncertainty-quantification and hypothesis-testing guarantees based on KL uncertainty sets, including applications to pessimistic value and Q-function estimation under model misspecification. Finally, we apply the same framework to universal source coding and obtain explicit coding error exponents for controlled Markov trajectories. Together, these results provide a unified type-theoretic approach to rare-event analysis, uncertainty quantification, offline reinforcement learning, and information-theoretic coding for finite controlled Markov systems.