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Due to copyright restrictions, the access to the full text of this article is only available via subscription. This chapter studies the finite-model approximation of discrete-time partially observed Markov decision process. We will find that by performing the standard reduction method, where one transforms a partially observed model to a belief-based fully observed model, we can apply and properly generalize the results in the preceding chapters to obtain approximation results. The versatility of approximation results under weak continuity conditions become particularly evident while investigating the applicability of these results to the partially observed case. We also provide systematic procedures for the quantization of the set of probability measures on the state space of POMDPs which is the state space of belief-MDPs.

Due to copyright restrictions, the access to the full text of this article is only available via subscription. In this chapter, we study the approximation of Witsenhausen’s counterexample and the Gaussian relay channel problem by using the results of the previous chapter. In particular, our goal is to establish that finite models obtained through the uniform quantization of the observation and action spaces result in a sequence of policies whose costs converge to the value function. We note that the operation of quantization has typically been the method to show that a non-linear policy can perform better than an optimal linear policy, both for Witsenhausen’s counterexample [10, 86] and the Gaussian relay channel problem [88, 152]. Our findings show that for a large class of problems, quantized policies not only may perform better than linear policies, but that they are actually almost optimal.