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

Abstract

Gradient information on the sampling distribution can be used to reduce the variance of Monte Carlo estimators via Stein's method. An important application is that of estimating an expectation of a test function along the sample path of a Markov chain, where gradient information enables convergence rate improvement at the cost of a linear system which must be solved. The contribution of this paper is to establish theoretical bounds on convergence rates for a class of estimators based on Stein's method. Our analysis accounts for (i) the degree of smoothness of the sampling distribution and test function, (ii) the dimension of the state space, and (iii) the case of non-independent samples arising from a Markov chain. These results provide insight into the rapid convergence of gradient-based estimators observed for low-dimensional problems, as well as clarifying a curse-of-dimension that appears inherent to such methods.

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

“…To the best of our knowledge, this optimality of Bayesian quadrature has not been established before, while recently there has been extensive theoretical analysis on Bayesian quadrature [8,9,44,4].…”

Variance Reduction in Monte Carlo Estimators via Empirical Variance Minimization

“…• As compared to control functionals (2) considered in Oates et al [15] and Oates et al [16], (14) includes partial derivatives of order greater than one. The usefulness of higher order partial derivatives can be demonstrated by the following example.…”

“…Alternatively, if an orthonormal system in L 2 (π) is analytically available, then one can build suitable control variates as finite sums of the corresponding basis functions. Furthermore, if π is known only up to a normalizing constant (which is often the case in Bayesian statistics), one can apply the recent approach of Oates et al [15] and Oates et al [16] suggesting the control variates which depend only on the ratio ∇π(x)/π(x). One way of constructing control variates ξ ∈ Ξ is based on consideration of measurable functions ζ φ of X satisfying E π [ζ φ ] = 0.…”

Variance reduction for Markov chains with application to MCMC

“…In this section we study numerical performance of the ESVM method for simulated and realworld data. Python implementation is available at https://github.com/svsamsonov/esvm Following Assaraf and Caffarel [2], Mira et al [25], Oates et al [29], we choose G to be a class of Stein control functionals of the form:…”

“…If ∇ log π is known, one can use popular zero-variance control variates based on the Stein's identity, see Assaraf and Caffarel [2] and Mira et al [25]. A non-parametric extension of such control variates is suggested in Oates et al [28] and Oates et al [27]. Control variates can be also obtained using the Poisson equation.…”