Algebraic Reduction of Hidden Markov Models
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866929392478846976 |
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| author | Grigoletto, Tommaso Ticozzi, Francesco |
| author_facet | Grigoletto, Tommaso Ticozzi, Francesco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_05968 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Algebraic Reduction of Hidden Markov Models Grigoletto, Tommaso Ticozzi, Francesco Machine Learning Optimization and Control Probability 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. |
| title | Algebraic Reduction of Hidden Markov Models |
| topic | Machine Learning Optimization and Control Probability |
| url | https://arxiv.org/abs/2208.05968 |