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

Oates, CJ; Cockayne, J; Briol, FX; Girolami, M; (2019) Convergence rates for a class of estimators based on Stein’s method. Bernoulli , 25 (2) pp. 1141-1159. 10.3150/17-BEJ1016. Green open access

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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.

Type: Article
Title: Convergence rates for a class of estimators based on Stein’s method
Open access status: An open access version is available from UCL Discovery
DOI: 10.3150/17-BEJ1016
Publisher version: https://doi.org/10.3150/17-BEJ1016
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: asymptotics; control functionals; reproducing kernel; scattered data; variance reduction
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/10079228
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