A Class of Solvable Markov Decision Models with Incomplete Information

This paper investigates natural conditions for the existence of optimal policies for a Markov decision process with incomplete information (MDPII) and with expected total costs. MDPII is the classic model of a controlled stochastic process with incomplete state observations which is more general than Partially Observable Markov Decision Processes (POMDPs). For MDPII we introduce the notion of a semi-uniform Feller transition probability, which is stronger than the notion of a weakly continuous transition probability. We show that an MDPII has a semi-uniform Feller transition probability if and only if the corresponding belief MDP also has a semi-uniform Feller transition probability. This fact has several corollaries. In particular, it provides new and implies all known sufficient conditions for the existence of optimal policies for POMDPs with expected total costs.