Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

Abstract

The Monte Carlo within Metropolis (MCwM) algorithm, interpreted as a perturbed Metropolis-Hastings (MH) algorithm, provides an approach for approximate sampling when the target distribution is intractable. Assuming the unperturbed Markov chain is geometrically ergodic, we show explicit estimates of the difference between the nth step distributions of the perturbed MCwM and the unperturbed MH chains. These bounds are based on novel perturbation results for Markov chains which are of interest beyond the MCwM setting. To apply the bounds, we need to control the difference between the transition probabilities of the two chains and to verify stability of the perturbed chain.

Stability of doubly-intractable distributions

“…We assume that for any θ ∈ Θ we can sample w.r. , as well as by [14,Lemma 23] we have for the second inverse moment of Q N (θ) that EQ N (θ) −2 1/2 ≤ EQ 1 (θ) −2 1/2 . Hence by Corollary 2.11 we obtain…”

“…The inexact setting leads to Markov chains which not necessarily target π Z , but another distribution that is (hopefully) close to π Z . In particular, the noisy Markov chain approach, for example investigated in [1,14,18] falls into this category. In addition to that also adaptive Markov chain Monte Carlo approaches have been developed, see [2,10].…”

Quasi-Stationary Monte Carlo and The Scale Algorithm

“…This is closely related to the perturbation theory of Markov chains; for details and recent advances see Hosseini and Johndrow (2019), Medina‐Aguayo et al . (2020), Mitrophanov (2005) and Rudolf and Schweizer (2018) and the references therein. In contrast, however, the present situation is more subtle, since we must work with conditional distributions and quasi ‐stationary distributions.…”

“…The paper introduces the ScaLE algorithm-a novel algorithm for sampling that is not based on the usual Metropolis accept-reject schemes. At its core, ScaLE is still an accept-reject sampler: for fixed running time T , one can obtain (approximate) samples from the distribution π by simulating a Brownian motion {X t } 0 t T This is closely related to the perturbation theory of Markov chains; for details and recent advances see Hosseini and Johndrow (2019), Medina-Aguayo et al (2020), Mitrophanov (2005) and Rudolf and Schweizer (2018) and the references therein. In contrast, however, the present situation is more subtle, since we must work with conditional distributions and quasi-stationary distributions.…”

Approximate Spectral Gaps for Markov Chain Mixing Times in High Dimensions

“…Another possible approach to bounding the mixing time of K using information from the restricted kernel is to bound directly (for instance, using coupling) the total variation distance between \pi 0 K N and \pi 0 (K \scrX 0 ) N . This has been explored in the literature [2,6,18], and more systematically by [24]. This approach typically works well when K has well-understood drift conditions.…”