A Dirichlet form approach to MCMC optimal scaling

Abstract

This paper shows how the theory of Dirichlet forms can be used to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions.2010 Mathematics Subject Classification: 60F05; 60J22, 65C05.

Counterexamples for optimal scaling of Metropolis–Hastings chains with rough target densities

“…This is enough to establish our objective, equation (35). And for this it suffices to show that the almost sure versions of (36) and (37) imply ESJD n (B (H ) , ϑ n ) → 0 almost surely.…”

“…Unfortunately, when ϑ n σ n → ∞ we can only show convergence in probability (35) ESJD n B (H ) , ϑ n…”

“…for an appropriate positive random variable θ that is B (H ) -measurable (see Theorem 1 or Theorem 2 and Roberts, Gelman and Gilks [29], Proposition 2.4). This can be shown by adopting the method of proof of Corollary 18 in [35], where we realise all the X i , Y (n) i on the same probability space and use the tower property.…”

“…To be specific, we plan to adapt the Dirichlet form methodology of Zanella, Bédard and Kendall [35] to deliver these anomalous scaling results at the level of weak convergence. With the same methodology we also expect to recover the MALA results of Roberts and Rosenthal [30] with smoothness assumptions only slightly stronger than C 3 (R).…”

“…These assumptions were necessitated by the methods of proof but did not otherwise seem particularly natural and it was unclear to what extent they were actually necessary. Recent work has used different methods of proof to establish, at least in the case of RWM, that the original smoothness assumptions were indeed much stricter than is really required [7,35]. The main focus of our paper is to develop a class of counterexamples to demonstrate the extent to which some kinds of smoothness assumption are genuinely necessary for both RWM and MALA.…”