Algebraic Reduction of Hidden Markov Models

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