Algebraic Reduction of Hidden Markov Models

The problem of reducing a Hidden Markov Model (HMM) to a one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach, adapted to HMMs by leveraging on a suitable algebraic representation of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that reproduce the single-time distribution of a given output process, while in the second the full (multi-time) distribution is preserved. The reduction method exploits not only the structure of the observed output, but also its initial condition, whenever the latter is known or belongs to a given subclass. Optimal algorithms are derived for a class of HMM, namely observable ones. In the general case, we propose algorithms that have produced minimal models for all the examples we analyzed, and conjecture their optimality.