Geometry and Duality of Alternating Markov Chains

TL;DR

This paper explores the geometric structure of alternating Markov chains by representing their half-steps as projections in probability space, revealing a duality between even and odd steps through information geometry.

Contribution

It introduces a novel geometric perspective on Markov chains using alternating projections and establishes a duality between different chain steps via Kullback-Leibler divergence.

Findings

Half-steps of Markov chains are realized as alternating projections.

A geometric proof of duality between even and odd chain steps.

Duality is characterized using reverse Kullback-Leibler divergence.

Abstract

In this note, we realize the half-steps of a general class of Markov chains as alternating projections with respect to the reverse Kullback-Leibler divergence between convex sets of joint probability distributions. Using this characterization, we provide a geometric proof of an information-theoretic duality between the Markov chains defined by the even and odd half-steps of the alternating projection scheme.