Minimax Testing of Identity to a Reference Ergodic Markov Chain

TL;DR

This paper introduces an efficient method for testing whether an unknown Markov chain matches a reference chain using a single long sequence, with sample complexity depending only on the reference chain's properties.

Contribution

It provides nearly optimal sample complexity bounds for identity testing of Markov chains, revealing that complexity depends solely on the reference chain, not the unknown one.

Findings

Sample complexity depends only on the reference chain's properties.

Achieves nearly matching upper and lower bounds.

Sample complexity is independent of the unknown chain's properties.

Abstract

We exhibit an efficient procedure for testing, based on a single long state sequence, whether an unknown Markov chain is identical to or $\varepsilon$-far from a given reference chain. We obtain nearly matching (up to logarithmic factors) upper and lower sample complexity bounds for our notion of distance, which is based on total variation. Perhaps surprisingly, we discover that the sample complexity depends solely on the properties of the known reference chain and does not involve the unknown chain at all, which is not even assumed to be ergodic.