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Characterization of Exponential Families of Lumpable Stochastic Matrices

Shun Watanabe, Geoffrey Wolfer

TL;DR

The paper resolves when lumpable stochastic matrices from a fixed lumping map form an exponential family. It introduces a dimension-based framework to certify $e$-family structure via the $e$-hull and a polynomial-time dimension test, complemented by combinatorial criteria involving merging blocks and lazy cycles. It shows both sufficient and necessary conditions (including a new dimensional criterion) and provides a full classification for very small state spaces, as well as algorithmic strategies for moderate sizes. The results advance understanding of the information-geometric structure of lumpable Markov models and yield practical tools for model selection and inference in reduced-state settings.

Abstract

It is known that the set of lumpable Markov chains over a finite state space, with respect to a fixed lumping function, generally does not form an exponential family of stochastic matrices. In this work, we explore efficiently verifiable necessary and sufficient conditions for families of lumpable transition matrices to form exponential families. To this end, we develop a broadly applicable dimension-based method for determining whether a given family of stochastic matrices forms an exponential family.

Characterization of Exponential Families of Lumpable Stochastic Matrices

TL;DR

The paper resolves when lumpable stochastic matrices from a fixed lumping map form an exponential family. It introduces a dimension-based framework to certify -family structure via the -hull and a polynomial-time dimension test, complemented by combinatorial criteria involving merging blocks and lazy cycles. It shows both sufficient and necessary conditions (including a new dimensional criterion) and provides a full classification for very small state spaces, as well as algorithmic strategies for moderate sizes. The results advance understanding of the information-geometric structure of lumpable Markov models and yield practical tools for model selection and inference in reduced-state settings.

Abstract

It is known that the set of lumpable Markov chains over a finite state space, with respect to a fixed lumping function, generally does not form an exponential family of stochastic matrices. In this work, we explore efficiently verifiable necessary and sufficient conditions for families of lumpable transition matrices to form exponential families. To this end, we develop a broadly applicable dimension-based method for determining whether a given family of stochastic matrices forms an exponential family.

Paper Structure

This paper contains 23 sections, 26 theorems, 135 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Lumpability
  5. Exponential families of stochastic matrices
  6. Foliation on the lumpable family.
  7. Problem statement---
  8. Motivation---
  9. First approach---
  10. The lumpable cone
  11. Characterizing of e-families of lumpable stochastic matrices
  12. Sufficient combinatorial conditions
  13. No multi-row merging block criterion
  14. Lazy-cycle criterion
  15. Necessary combinatorial conditions
  16. ...and 8 more sections

Key Result

Proposition 2.1

For any fixed $P^\flat_0 \in \mathcal{W}(\mathcal{X}, \mathcal{D})$,

Figures (1)

  • Figure 1: We can regard $\mathcal{W}_\kappa(\mathcal{Y}, \mathcal{E})$ as a section of the cone $\mathcal{F}^+_\kappa(\mathcal{Y}, \mathcal{E})$.

Theorems & Definitions (69)

  • Definition 2.1: ${\mathfrak{s}}$-normalization
  • Definition 2.2: nagaoka2005exponential
  • Definition 2.3: e-family of stochastic matrices nagaoka2005exponential
  • Remark 2.1
  • Example 2.1: Birth-and-death-chains
  • Proposition 2.1: Foliation on $\mathcal{W}_{\kappa}(\mathcal{Y}, \mathcal{E})$ wolfer2024geometric
  • Proposition 3.1: Commutativity
  • proof
  • Proposition 3.2: Closure under similarity transform
  • proof
  • ...and 59 more