Minimal Markov Models

TL;DR

This paper introduces a new class of finite order Markov chain models that efficiently identify the minimal set of parameters needed to represent a source, using an equivalence relation and Bayesian information criterion for model selection.

Contribution

It proposes a novel approach to minimal Markov models by defining an equivalence relation on states and demonstrates consistent model selection with BIC.

Findings

Model can be selected consistently using BIC.

Defines an equivalence relation to reduce parameters.

Provides a richer class of finite order Markov models.

Abstract

In this work we introduce a new and richer class of finite order Markov chain models and address the following model selection problem: find the Markov model with the minimal set of parameters (minimal Markov model) which is necessary to represent a source as a Markov chain of finite order. Let us call $M$ the order of the chain and $A$ the finite alphabet, to determine the minimal Markov model, we define an equivalence relation on the state space $A^{M}$, such that all the sequences of size $M$ with the same transition probabilities are put in the same category. In this way we have one set of $(|A|-1)$ transition probabilities for each category, obtaining a model with a minimal number of parameters. We show that the model can be selected consistently using the Bayesian information criterion.