Approximation of Markov Processes by Lower Dimensional Processes

In this paper, we investigate the problem of aggregating a given finite-state Markov process by another process with fewer states. The aggregation utilizes total variation distance as a measure of discriminating the Markov process by the aggregate process, and aims to maximize the entropy of the aggregate process invariant probability, subject to a fidelity described by the total variation distance ball. An iterative algorithm is presented to compute the invariant distribution of the aggregate process, as a function of the invariant distribution of the Markov process. It turns out that the approximation method via aggregation leads to an optimal aggregate process which is a hidden Markov process, and the optimal solution exhibits a water-filling behavior. Finally, the algorithm is applied to specific examples to illustrate the methodology and properties of the approximations.