Shared Information for a Markov Chain on a Tree

TL;DR

This paper provides a new proof for characterizing shared information in Markov chains on trees, introduces a global Markov property based on graph separation, and develops an algorithm for estimating shared information when the distribution is unknown.

Contribution

It offers a novel proof of shared information characterization, establishes a global Markov property for tree-structured Markov chains, and proposes an estimation algorithm with error analysis.

Findings

Explicit characterization of shared information in Markov chains on trees

Global Markov property based on graph separation

Multiarmed bandit algorithm for estimating shared information

Abstract

Shared information is a measure of mutual dependence among multiple jointly distributed random variables with finite alphabets. For a Markov chain on a tree with a given joint distribution, we give a new proof of an explicit characterization of shared information. The Markov chain on a tree is shown to possess a global Markov property based on graph separation; this property plays a key role in our proofs. When the underlying joint distribution is not known, we exploit the special form of this characterization to provide a multiarmed bandit algorithm for estimating shared information, and analyze its error performance.