Sequential One-Sided Hypothesis Testing of Markov Chains

TL;DR

This paper develops a sequential hypothesis testing method for Markov chains that adapts to unknown alternatives using data-driven estimators, achieving near-optimal performance without prior knowledge.

Contribution

It introduces a family of SPRT-type tests for Markov chains that adaptively estimate the alternative distribution, matching SPRT performance asymptotically.

Findings

Test performance matches SPRT in simple cases

Estimator with $ ext{O}( ext{log} t)$ regret guarantees effective adaptation

Method applicable to various engineering problems involving Markov processes

Abstract

We study the problem of sequentially testing whether a given stochastic process is generated by a known Markov chain. Formally, given access to a stream of random variables, we want to quickly determine whether this sequence is a trajectory of a Markov chain with a known transition matrix $P$ (null hypothesis) or not (composite alternative hypothesis). This problem naturally arises in many engineering problems. The main technical challenge is to develop a sequential testing scheme that adapts its sample size to the unknown alternative. Indeed, if we knew the alternative distribution (that is, the transition matrix) $Q$, a natural approach would be to use a generalization of Wald's sequential probability ratio test (SPRT). Building on this intuition, we propose and analyze a family of one-sided SPRT-type tests for our problem that use a data-driven estimator $\hat{Q}$. In particular, we show that if the deployed estimator admits a worst-case regret guarantee scaling as $\mathcal{O}\left( \log{t} \right)$, then the performance of our test asymptotically matches that of SPRT in the simple hypothesis testing case. In other words, our test automatically adapts to the unknown hardness of the problem, without any prior information. We end with a discussion of known Markov chain estimators with $\mathcal{O}\left( \log{t} \right)$ regret.