The space of positive transition measures on a Markov chain

TL;DR

The paper addresses how to assign a statistically meaningful, dually flat information-geometric structure to the space of Markov-chain transitions by extending Amari's positive-measure framework. It constructs an extended space of positive edge-weights $\mathcal{F}^+$ and an extended parameter space $\overline{M}$, proving a single $F$-divergence that is also a Bregman divergence and recovers the Nagaoka geometry when restricted to $\mathcal{W}$. By introducing a hyperplane slice $\tilde{M}$ with $r(\eta)=1$, it defines a genuine dually flat manifold $\tilde{\mathcal{W}}$ of positive transition measures, thereby generalizing the canonical divergence and embedding geometry to a broader setting. The work also outlines potential statistical applications, including estimation and hypothesis testing for Markov models via $F$-divergences and robust projection-based methods, with implications for THMC models and Markov-embedding theory.

Abstract

Information geometry of Markov chains has been studied using the dually flat structure of the space of transition probabilities. Although applications of this structure have been investigated, few attempts have examined its statistical meaning. In this paper, we construct a foundation for investigating the statistical meaning based on Amari's theory of positive measures. For the space of discrete distributions, Amari has introduced the space of positive measures by removing the constraint condition and investigated the extended space by finding the Bregman and $F$-divergence suitably. According to this, we introduce an extension of the space of transition probabilities equipped with suitable $F$-divergence for a given Markov chain. We regard it as the space of positive transition measures on a Markov chain, and study its dually flat structure. This provides new insight into the geometry of Markov chains and may lead to the development of the theory of Markov embeddings.