Information Geometry of Reversible Markov Chains

TL;DR

This paper explores the geometric structure of reversible Markov chains, characterizing their exponential and mixture family properties, and introduces new parametrizations, projections, and notions of reversiblization within information geometry.

Contribution

It provides a comprehensive geometric analysis of reversible Markov kernels, including new parametrizations, projections, and insights into their exponential family structure.

Findings

Reversible Markov kernels form both exponential and mixture families.

Closed-form expressions for information projections onto the reversible manifold.

Reversible kernels are the minimal exponential family generated by symmetric kernels.

Abstract

We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels.