Information geometry of Markov chains has been studied by Nagaoka, Takeuchi and others using the dually flat structure of the space of transition probabilities. In this context, a submanifold of the space is called a Markov model. In the present paper, we seek for a theory of extended spaces of Markov models in the following sense. As a prototype, for the space of probability distributions on a finite set, Amari has introduced the space of positive measures simply by removing the constraint condition that the total mass is equal to $1$ and investigated the extended space by finding the Bregman and $F$-divergence suitably. According to this line, 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 the dually flat structure on the space. That provides a new insight on the geometry of Markov chains. We also discuss a relation with other existing work.

翻译：关于马尔可夫链的信息几何，长冈、竹内等人利用转移概率空间的对偶平坦结构进行了研究。在此背景下，该空间的子流形被称为马尔可夫模型。本文旨在从以下角度探索马尔可夫模型扩展空间的理论。作为原型，对于有限集上的概率分布空间，甘利俊一通过移除总质量等于1的约束条件引入了正测度空间，并通过适当确定布雷格曼散度和$F$-散度对扩展空间进行了研究。遵循这一思路，我们针对给定的马尔可夫链，引入配备了适当$F$-散度的转移概率空间的扩展。我们将其视为马尔可夫链上的正转移测度空间，并研究该空间的对偶平坦结构。这为马尔可夫链的几何提供了新的见解。此外，我们还讨论了与现有其他工作的关系。