Testing Symmetric Markov Chains from a Single Trajectory

Classical distribution testing assumes access to i.i.d. samples from the distribution that is being tested. We initiate the study of Markov chain testing, assuming access to a single trajectory of a Markov Chain. In particular, we observe a single trajectory X0, . . . , Xt, . . . of an unknown, symmetric, and finite state Markov ChainM. We do not control the starting state X0, and we cannot restart the chain. Given our single trajectory, the goal is to test whether M is identical to a model Markov Chain M′, or far from it under an appropriate notion of difference. We propose a measure of difference between two Markov chains, motivated by the early work of Kazakos [Kaz78], which captures the scaling behavior of the total variation distance between trajectories sampled from the Markov chains as the length of these trajectories grows. We provide efficient testers and information-theoretic lower bounds for testing identity of symmetric Markov chains under our proposed measure of difference, which are tight up to logarithmic factors if the hitting times of the model chain M′ is Õ(n) in the size of the state space n. Supported by a Microsoft Research Faculty Fellowship, and NSF Award CCF-1551875, CCF-1617730 and CCF1650733. Supported by NSF Award CCF-1551875, CCF-1617730 and CCF-1650733. Supported by NSF Award CCF-1551875, CCF-1617730 and CCF-1650733. ar X iv :1 70 4. 06 85 0v 2 [ cs .L G ] 4 D ec 2 01 7