Algebraic Reduction of Hidden Markov Models

TL;DR

This paper introduces algebraic and system-theoretic methods to reduce Hidden Markov Models to smaller, equivalent models that preserve specific probabilistic distributions, enhancing model efficiency.

Contribution

It extends realization theory to HMMs and proposes two algorithms for exact model reduction using algebraic representations and stochastic projections.

Findings

Algorithms preserve single-time and multi-time distributions.

Reduction exploits initial conditions for improved accuracy.

Optimal algorithms are developed for observable HMMs.

Abstract

The problem of reducing a Hidden Markov Model (HMM) to one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach. Realization theory tools are extended to HMMs by leveraging suitable algebraic representations of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that exactly 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.