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Local Equivalence Problem in Hidden Markov Model

Masahito Hayashi

Abstract

In the hidden Markov process, there is a possibility that two different transition matrices for hidden and observed variables yield the same stochastic behavior for the observed variables. Since such two transition matrices cannot be distinguished, we need to identify them and consider that they are equivalent, in practice. We address the equivalence problem of hidden Markov process in a local neighborhood by using the geometrical structure of hidden Markov process. For this aim, we introduce a mathematical concept to express Markov process, and formulate its exponential family by using generators. Then, the above equivalence problem is formulated as the equivalence problem of generators. Taking this equivalence problem into account, we derive several concrete parametrizations in several natural cases.

Local Equivalence Problem in Hidden Markov Model

Abstract

In the hidden Markov process, there is a possibility that two different transition matrices for hidden and observed variables yield the same stochastic behavior for the observed variables. Since such two transition matrices cannot be distinguished, we need to identify them and consider that they are equivalent, in practice. We address the equivalence problem of hidden Markov process in a local neighborhood by using the geometrical structure of hidden Markov process. For this aim, we introduce a mathematical concept to express Markov process, and formulate its exponential family by using generators. Then, the above equivalence problem is formulated as the equivalence problem of generators. Taking this equivalence problem into account, we derive several concrete parametrizations in several natural cases.

Paper Structure

This paper contains 33 sections, 29 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of obtained results
  3. Global equivalence
  4. Local equivalence
  5. Outline of remaining parts
  6. Hidden Markov model and equivalence
  7. Notations with ${\cal Y}$-indexed transition matrix
  8. Equivalence relation
  9. Exponential family of ${\cal Y}$-indexed transition matrices
  10. Definition of exponential family
  11. Local equivalence
  12. Construction of linearly independent generators
  13. Two-hidden-state case in general model
  14. Two-state observation case
  15. Non-singular points
  16. ...and 18 more sections

Key Result

theorem 1

The following conditions for two collections of non-negative matrices $\vec{W}$, $\vec{W}'$ and two distributions $P$, $P'$ on ${\cal X}$ are equivalent.

Figures (3)

  • Figure 1: The second model: The transition matrix $W$ determines the Markov process on the set ${\cal X}$ of hidden states. The transition matrix $V$ determines the observed variable $Y$ with the condition on the hidden variable $X$.
  • Figure 2: The first model: The transition matrix $W$ determines the Markov process on the set ${\cal X}$ of hidden states. The function $f$ of the hidden variable $X$ determines the observed variable $Y$.
  • Figure 3: The third model: the hidden variable $X_i$ and the observed variable $Y_i$ are correlated even when the previous hidden variable $X_{i-1}$is fixed.

Theorems & Definitions (49)

  • theorem 1
  • theorem 2
  • theorem 3
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • ...and 39 more