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Algebraic Reduction of Hidden Markov Models

Tommaso Grigoletto, Francesco Ticozzi

TL;DR

The paper addresses the problem of reducing Hidden Markov Models to a smaller dimension while exactly preserving output marginals. It introduces an algebraic probability framework and a system-theoretic viewpoint to construct reduced HMMs via stochastic projections and conditional expectation factorization. Two reduction problems are analyzed: single-time marginals and multi-time marginals, with algorithms that yield reduced HMMs that reproduce the original marginals and, in the multi-time case, the full sequence probabilities; optimality results are provided for observable HMMs and guidance on choosing the effective subspace. The approach enables exact coarse-graining, highlights the role of initial conditions in minimal reductions, and suggests future extensions to approximate reductions and quantum settings.

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.

Algebraic Reduction of Hidden Markov Models

TL;DR

The paper addresses the problem of reducing Hidden Markov Models to a smaller dimension while exactly preserving output marginals. It introduces an algebraic probability framework and a system-theoretic viewpoint to construct reduced HMMs via stochastic projections and conditional expectation factorization. Two reduction problems are analyzed: single-time marginals and multi-time marginals, with algorithms that yield reduced HMMs that reproduce the original marginals and, in the multi-time case, the full sequence probabilities; optimality results are provided for observable HMMs and guidance on choosing the effective subspace. The approach enables exact coarse-graining, highlights the role of initial conditions in minimal reductions, and suggests future extensions to approximate reductions and quantum settings.

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.
Paper Structure (24 sections, 20 theorems, 72 equations, 1 figure, 2 algorithms)

This paper contains 24 sections, 20 theorems, 72 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Basic Notation
  3. Algebraic approach to probability theory
  4. Fundamentals of algebraic probabilistic models
  5. Events and $\sigma$-Algebras
  6. Random variables
  7. Probability and expectations
  8. Stochastic maps and Conditional Expectations
  9. HMM and Problem definition
  10. Preliminary results: A system theoretic viewpoint
  11. Single time-marginal problem
  12. Multi-time marginal problem
  13. Single-time solution
  14. Intuition
  15. Proposed solution
  16. ...and 9 more sections

Key Result

Proposition 1

If $\mathcal{F}\subset \mathbb{R}^n$ is a vector $\sigma$-algebra, then $\mathscr{A} = {\rm span}\{\mathcal{F}\}$ is the smallest subalgebra in $\mathbb{R}^n$ containing $\mathcal{F}$, and it is unital. Conversely, let $\mathscr{A}$ be any unital subalgebra in $\mathbb{R}^n$ and ${\rm idem}(\mathscr

Theorems & Definitions (55)

  • Proposition 1
  • Lemma 1
  • Remark 1
  • Proposition 2
  • proof
  • Corollary 1
  • proof
  • Definition 1: Hidden Markov processes
  • Definition 2: Hidden Markov models
  • Remark 2
  • ...and 45 more