Let M be a reversible Markov chain, with stationary distribution Q. Suppose we construct a metropolis Hastings proposal in order to get a chain N with stationary distribution P.
I'd like to bound the mixing time of N by some quantity related to the mixing time of M, and the total variation distance between P and Q. (Or similar.)
Can someone provide me with a reference?
Edit: I've been told that essentially no bound exists. The reason is because even if the distributions of P and Q are similar on most of the graph, they could differ dramatically on some nodes that are crucial to the mixing of P. (E.g. if P is zero along a thin set of nodes that disconnect the graph associated to Q...)