Random perturbations of stochastic chains with unbounded variable length memory

TL;DR

This paper studies how small random noise affects infinite order stochastic chains with variable length memory, demonstrating that the original context structure can be recovered despite perturbations.

Contribution

It introduces a method to recover the context tree of unbounded variable length memory chains under small random perturbations using a modified Context algorithm.

Findings

Transition probabilities remain close under noise

Context tree can be recovered with small noise

Algorithm adapts to unbounded variable length memory

Abstract

We consider binary infinite order stochastic chains perturbed by a random noise. This means that at each time step, the value assumed by the chain can be randomly and independently flipped with a small fixed probability. We show that the transition probabilities of the perturbed chain are uniformly close to the corresponding transition probabilities of the original chain. As a consequence, in the case of stochastic chains with unbounded but otherwise finite variable length memory, we show that it is possible to recover the context tree of the original chain, using a suitable version of the algorithm Context, provided that the noise is small enough.