  We present an alternative view for the study of optimal control of partially
observed Markov Decision Processes (POMDPs). We first revisit the traditional
(and by now standard) separated-design method of reducing the problem to fully
observed MDPs (belief-MDPs), and present conditions for the existence of
optimal policies. Then, rather than working with this standard method, we
define a Markov chain taking values in an infinite dimensional product space
with control actions and the state process causally conditionally independent
given the measurement/information process. We provide new sufficient conditions
for the existence of optimal control policies under the discounted cost and
average cost infinite horizon criteria. In particular, while in the belief-MDP
reduction of POMDPs, weak Feller condition requirement imposes total variation
continuity on either the system kernel or the measurement kernel, with the
approach of this paper only weak continuity of both the transition kernel and
the measurement kernel is needed (and total variation continuity is not)
together with regularity conditions related to filter stability. For the
discounted cost setup, we establish near optimality of finite window policies
via a direct argument involving near optimality of quantized approximations for
MDPs under weak Feller continuity, where finite truncations of memory can be
viewed as quantizations of infinite memory with a uniform diameter in each
finite window restriction under the product metric. For the average cost setup,
we provide new existence conditions and also a general approach on how to
initialize the randomness which we show to establish convergence to optimal
cost. In the control-free case, our analysis leads to new and weak conditions
for the existence and uniqueness of invariant probability measures for
non-linear filter processes.
