Comparing entropy rates on finite and infinite rooted trees

TL;DR

This paper compares entropy rates of stochastic processes on finite and infinite rooted trees, extending existing methods to trees with infinite paths and countably many successors, using Kullback-Leibler divergence.

Contribution

It introduces new techniques for comparing entropy rates of processes on complex trees, including infinite and countably branching trees, with applications to perturbations.

Findings

Extended entropy comparison methods to infinite trees

Analyzed processes with countably many successors

Studied perturbations on infinite geodesic trees

Abstract

We consider stochastic processes with (or without) memory whose evolution is encoded by a finite or infinite rooted tree. The main goal is to compare the entropy rates of a given base process and a second one, to be considered as a perturbation of the former. The processes are described by probability measures on the boundary of the given tree, and by corresponding forward transition probabilities at the inner nodes. The comparison is in terms of Kullback-Leibler divergence. We elaborate and extend ideas and results of B\"ocherer and Amjad. Our extensions involve length functions on the edges of the tree as well as nodes with countably many successors. In particular, in the last part, we consider trees with infinite geodesic rays and random perturbations of a given process.