Convergence of Probability Measures and Markov Decision Models with Incomplete Information

TL;DR

This paper analyzes different types of convergence of probability measures on metric spaces, providing criteria and conditions, and applies these results to control problems in Partially Observable Markov Decision Processes with incomplete information.

Contribution

It offers new criteria for convergence of probability measures and stochastic kernels, and applies these to control models with incomplete information.

Findings

Criteria for weak and setwise convergence established

Continuity conditions for stochastic kernels derived

Applications to control of Partially Observable Markov Decision Processes

Abstract

This paper deals with three major types of convergence of probability measures on metric spaces: weak convergence, setwise converges, and convergence in the total variation. First, it describes and compares necessary and sufficient conditions for these types of convergence, some of which are well-known, in terms of convergence of probabilities of open and closed sets and, for the probabilities on the real line, in terms of convergence of distribution functions. Second, it provides % convenient criteria for weak and setwise convergence of probability measures and continuity of stochastic kernels in terms of convergence of probabilities defined on the base of the topology generated by the metric. Third, it provides applications to control of Partially Observable Markov Decision Processes and, in particular, to Markov Decision Models with incomplete information.