  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.
