Develop a multiterminal Markovian analysis using the geometric decomposition (m/e-foliation) of multivariate chains

Develop a comprehensive analysis in a multiterminal Markovian setting that leverages the geometric decomposition of multivariate Markov chains introduced in the paper—namely, the exponential family of product chains and the mixture families of chains with prescribed marginal edge measures—to formulate and study multiterminal problems (e.g., hypothesis testing), analogous to the distributional setting of Amari’s decomposition and Watanabe’s multiterminal hypothesis testing.

Background

The paper establishes that product transition matrices form an exponential family and that multivariate transition matrices with prescribed marginal edge measures form mixture families, yielding a mutually dual (m/e) foliation and associated Pythagorean identities for Markov chains.

In distributional information geometry, similar e/m-decompositions underpin analyses of multiterminal hypothesis testing. Extending these tools to Markov chains could enable new results in multiterminal Markovian inference and performance bounds, but the authors do not carry out this development here.