A canonical representation for aggregated Markov processes

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.