Stochastically Perturbed Chains of Variable Memory

TL;DR

This paper investigates inference for variable memory chains under contamination, showing bounds on transition probability differences and methods to recover the original context tree despite contamination.

Contribution

It introduces bounds on transition probabilities under contamination and demonstrates how to recover the original context tree using a modified Context algorithm.

Findings

Transition probabilities can be uniformly bounded despite contamination.

The context tree can be recovered if contamination probability is sufficiently small.

The study provides theoretical guarantees for inference under contamination regimes.

Abstract

In this paper, we study inference for chains of variable order under two distinct contamination regimes. Consider we have a chain of variable memory on a finite alphabet containing zero. At each instant of time an independent coin is flipped and if it turns head a contamination occurs. In the first regime a zero is read independent of the value of the chain. In the second regime, the value of another chain of variable memory is observed instead of the original one. Our results state that the difference between the transition probabilities of the original process and the corresponding ones of the contaminated process may be bounded above uniformly. Moreover, if the contamination probability is small enough, using a version of the Context algorithm we are able to recover the context tree of the original process through a contaminated sample.