Markov Decision Processes with Incomplete Information and Semi-Uniform Feller Transition Probabilities

TL;DR

This paper introduces a class of partially observable stochastic control models with semi-uniform Feller transition probabilities, establishing conditions for optimal policies, value iteration convergence, and generalizing existing results in POMDPs.

Contribution

It defines and analyzes Markov Decision Processes with incomplete information and semi-uniform Feller transitions, extending theoretical foundations and ensuring the existence of optimal policies.

Findings

Optimal policies exist under mild conditions.

Value iteration converges to optimal values.

Generalizes conditions for weak continuity in POMDPs.

Abstract

This paper deals with control of partially observable discrete-time stochastic systems. It introduces and studies Markov Decision Processes with Incomplete Information and with semi-uniform Feller transition probabilities. The important feature of these models is that their classic reduction to Completely Observable Markov Decision Processes with belief states preserves semi-uniform Feller continuity of transition probabilities. Under mild assumptions on cost functions, optimal policies exist, optimality equations hold, and value iterations converge to optimal values for these models. In particular, for Partially Observable Markov Decision Processes the results of this paper imply new and generalize several known sufficient conditions on transition and observation probabilities for weak continuity of transition probabilities for Markov Decision Processes with belief states, the existence of optimal policies, validity of optimality equations defining optimal policies, and convergence of value iterations to optimal values.