On the Poisson equation for Metropolis-Hastings chains

Convergence rates for a class of estimators based on Stein’s method

“…This idea is motivated by the observation that the methods listed above are (in effect) solving a misspecified regression problem, since in general f does not belong to the linear span of the statistics {ψ i } k i=1 . The recent work by Mijatović and Vogrinc (2015); Oates et al (2017) alleviates model misspecification by increasing the number k of statistics alongside the number n of samples so that the limiting space spanned by the statistics {ψ i } ∞ i=1 is dense in a class of functions that contains the test function f of interest. Both methods provide a non-parametric alternative to classical control variates whose error is o P (n − 1 2 ).…”

“…Both methods provide a non-parametric alternative to classical control variates whose error is o P (n − 1 2 ). Of these two proposed solutions, Mijatović and Vogrinc (2015) is not considered here since it is unclear how to proceed when Π is known only up to a normalisation constant. On the other hand the control functional method of Oates et al (2017) is straight-forward to implement when gradients {∇ log π(x i )} n i=1 are provided.…”

Asymptotic variance for random walk Metropolis chains in high dimensions: logarithmic growth via the Poisson equation

“…Following ideas from [MV16], we construct and analyse the estimator using (I) and (II). Specifically, let ρ be a density on R and ρ d px d q :" ś d i"1 ρpx d i q the corresponding d-dimensional product density, where x d " px d 1 , .…”

“…For large d, the function f ought to approximate the solution of the Poisson equation for X d . The reasoning in [MV16] then suggests the form of an estimator for ρ d pf q, which under appropriate technical assumptions, satisfies the following CLT:…”

“…Using an approximate solution to the Poisson equation to construct control variates is a common variance reduction technique, see e.g. [Hen97,Mey08,DK12,MV16]. In this context, more often than not, an approximate solution used is a solution of a Poisson equation of a simpler related process.…”