Hidden Markov Models with Multiple Observation Processes

We consider a hidden Markov model with multiple observation processes, one of which is chosen at each point in time by a policy---a deterministic function of the information state---and attempt to determine which policy minimises the limiting expected entropy of the information state. Focusing on a special case, we prove analytically that the information state always converges in distribution, and derive a formula for the limiting entropy which can be used for calculations with high precision. Using this fomula, we find computationally that the optimal policy is always a threshold policy, allowing it to be easily found. We also find that the greedy policy is almost optimal.