Block Markov Chains on Trees

TL;DR

This paper introduces block Markov chains on trees, exploring their structure and relationship with Markov chains and Markov random fields, revealing new subclasses and properties of these processes.

Contribution

It defines block Markov chains on trees, characterizes their relation to Markov chains and MRFs, and identifies conditions under which BMCs coincide with or differ from these models.

Findings

BMCs form a strict subclass of MRFs on trees.

BMCs coincide with MCs in the one-dimensional case.

Perturbations can produce BMCs that are not MCs.

Abstract

We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMCs define a new class of tree-indexed Markovian processes. We clarify the structure of BMCs in connection with Markov chains (MCs) and Markov random fields (MRFs). Mainly, show that probability measures which are BMCs for every root are indeed Markov chains (MCs) and yet they form a strict subclass of Markov random fields (MRFs) on the considered tree. Conversely, a class of MCs which are BMCs is characterized. Furthermore, we establish that in the one-dimensional case the class of BMCs coincides with MCs. However, a slight perturbation of the one-dimensional lattice leads to us to an example of BMCs which are not MCs appear.