Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability

TL;DR

This paper characterizes when lumping a finite-state Markov chain preserves entropy rate, using combinatorial criteria based on the transition graph, and explores conditions for strong k-lumpability and entropy preservation.

Contribution

It provides a combinatorial characterization of entropy rate preservation and strong k-lumpability for Markov chain lumpings, extending understanding of higher-order lumpability.

Findings

Entropy rate preservation linked to growth rate of preimages.

Strong k-lumpability characterized by stationary entropic bounds.

Sufficient conditions for entropy preservation and k-lumpability in sparse settings.

Abstract

A lumping of a Markov chain is a coordinate-wise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, iff the lumped process is a k-th order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.