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arXiv · 2208.05968

Algebraic Reduction of Hidden Markov Models

Tommaso Grigoletto·Francesco Ticozzi
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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.

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Tommaso Grigoletto, Francesco Ticozzi. 2022-08-11. Algebraic Reduction of Hidden Markov Models. https://doi.org/10.1109/tac.2023.3279209

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