On estimating the memory for finitarily Markovian processes

TL;DR

This paper thoroughly investigates methods for universally estimating the finite memory length of finitarily Markovian processes, considering both past and future observation scenarios across finite and countably infinite alphabets.

Contribution

It provides a comprehensive analysis of the problem of estimating the memory length in finitarily Markovian processes, including both backward and forward estimation methods.

Findings

Complete characterization of the estimation problem

Analysis applicable to finite and countably infinite alphabets

Results on the feasibility of universal estimation

Abstract

Finitarily Markovian processes are those processes $\{X_n\}_{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K = K(\{X_n\}_{n=-\infty}^0$) such that the conditional distribution of $X_1$ given the entire past is equal to the conditional distribution of $X_1$ given only $\{X_n\}_{n=1-K}^0$. The least such value of $K$ is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of $K$, both in the backward sense that we have just described and in the forward sense, where one observes successive values of $\{X_n\}$ for $n \geq 0$ and asks for the least value $K$ such that the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=n-K+1}^n$ is the same as the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=-\infty}^n$. We allow for finite or countably infinite alphabet size.