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.

💡 Research Summary

The paper investigates a hidden Markov model (HMM) in which several distinct observation processes are available, but only one may be employed at each time step. The choice of observation is governed by a deterministic policy that maps the current information state—a sufficient statistic summarising all past observations—into an index indicating which observation process to use. The objective is to minimise the long‑run expected entropy of the information state, which serves as a proxy for the uncertainty about the underlying Markov chain.

The authors first formalise the extended HMM. A standard finite‑state, time‑homogeneous Markov chain (X_t) is coupled with a finite collection of observation processes ({Y^{(i)}t}{i\in\mathcal O}). An additional random index sequence (I_t) selects which observation is actually realised, yielding the actual observation (Y_t = Y^{(I_t)}_t). The information state (\zeta_t) is defined as the posterior distribution of (X_t) given the history of actual observations up to time (t-1). Updating (\zeta_t) follows the usual Bayesian recursion, which depends on the chosen observation matrix at each step.

A central theoretical contribution is the proof that, under the mild condition that each observation process provides perfect information with positive probability, the information state converges in distribution for almost every underlying Markov chain. In other words, the sequence ({\zeta_t}) possesses a unique stationary distribution (\zeta^*) that does not depend on the initial belief. This result guarantees that the limiting expected entropy \