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The space of positive transition measures on a Markov chain

Abstract

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 11 and investigated the extended space by finding the Bregman and FF-divergence suitably. According to this line, we introduce an extension of the space of transition probabilities equipped with suitable FF-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.

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