  We consider the problem of testing the identity of a reversible Markov chain
against a reference from a single trajectory of observations. Employing the
recently introduced notion of a lumping-congruent Markov embedding, we show
that, at least in a mildly restricted setting, testing identity to a reversible
chain reduces to testing to a symmetric chain over a larger state space and
recover state-of-the-art sample complexity for the problem.
