Information-Preserving Markov Aggregation

TL;DR

This paper introduces conditions and algorithms for reducing the state space of a Markov chain without losing information, enabling lossless compression while preserving the chain's entropy rate.

Contribution

It provides a sufficient condition for information-preserving state space reduction and an algorithm to find all such reductions for a given Markov chain.

Findings

The reduced state space's size is bounded by node degrees of the transition graph.

The algorithm can identify all possible lossless reductions.

Application demonstrated on an English text bi-gram model.

Abstract

We present a sufficient condition for a non-injective function of a Markov chain to be a second-order Markov chain with the same entropy rate as the original chain. This permits an information-preserving state space reduction by merging states or, equivalently, lossless compression of a Markov source on a sample-by-sample basis. The cardinality of the reduced state space is bounded from below by the node degrees of the transition graph associated with the original Markov chain. We also present an algorithm listing all possible information-preserving state space reductions, for a given transition graph. We illustrate our results by applying the algorithm to a bi-gram letter model of an English text.