Testing Markov Chains without Hitting
This addresses a fundamental problem in statistical inference for Markov chains, with potential applications in fields like machine learning and data analysis, though it appears incremental as it builds on prior work to remove a specific bottleneck.
The paper tackles the problem of identity testing for Markov chains, where the goal is to determine if an unknown transition matrix equals a known one or is far from it, using a single trajectory. The result is an algorithm that avoids dependence on hitting time, enabling efficient testing even when observing every state is infeasible.
We study the problem of identity testing of markov chains. In this setting, we are given access to a single trajectory from a markov chain with unknown transition matrix $Q$ and the goal is to determine whether $Q = P$ for some known matrix $P$ or $\text{Dist}(P, Q) \geq ε$ where $\text{Dist}$ is suitably defined. In recent work by Daskalakis, Dikkala and Gravin, 2018, it was shown that it is possible to distinguish between the two cases provided the length of the observed trajectory is at least super-linear in the hitting time of $P$ which may be arbitrarily large. In this paper, we propose an algorithm that avoids this dependence on hitting time thus enabling efficient testing of markov chains even in cases where it is infeasible to observe every state in the chain. Our algorithm is based on combining classical ideas from approximation algorithms with techniques for the spectral analysis of markov chains.