Algebraic Reduction of Hidden Markov Models

• URL: http://arxiv.org/abs/2208.05968v2
• Date: Fri, 19 May 2023 09:43:26 GMT
• Title: Algebraic Reduction of Hidden Markov Models
• Authors: Tommaso Grigoletto and Francesco Ticozzi
• Abstract summary: We propose two algorithms that return models that exactly reproduce the single-time distribution of a given output process. The reduction method exploits not only the structure of the observed output, but also its initial condition. Optimal algorithms are derived for a class of HMM, namely ones.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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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