Convergence rates for a class of estimators based on Stein’s method
Convergence rates for a class of estimators based on Stein’s method
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.
1141-1159
Oates, Chris J.
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Cockayne, Jonathan
da87c8b2-fafb-4856-938d-50be8f0e4a5b
Briol, Francois-Xavier
9fc3f5c5-ded6-402c-af8f-5ebcfe6bee89
Girolami, Mark
4feb7248-7beb-4edc-8509-139b4049d23b
1 May 2019
Oates, Chris J.
3af13c56-dc47-4d2c-867f-e4e933e74619
Cockayne, Jonathan
da87c8b2-fafb-4856-938d-50be8f0e4a5b
Briol, Francois-Xavier
9fc3f5c5-ded6-402c-af8f-5ebcfe6bee89
Girolami, Mark
4feb7248-7beb-4edc-8509-139b4049d23b
Oates, Chris J., Cockayne, Jonathan, Briol, Francois-Xavier and Girolami, Mark
(2019)
Convergence rates for a class of estimators based on Stein’s method.
Bernoulli, 25 (2), .
(doi:10.3150/17-bej1016).
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.
Text
Bernoulli_1551862846
- Accepted Manuscript
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Accepted/In Press date: 1 August 2017
Published date: 1 May 2019
Identifiers
Local EPrints ID: 482229
URI: http://eprints.soton.ac.uk/id/eprint/482229
ISSN: 1350-7265
PURE UUID: d764a942-d91a-4c81-95cc-36c2aead36fa
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Date deposited: 21 Sep 2023 16:53
Last modified: 17 Mar 2024 04:09
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Author:
Chris J. Oates
Author:
Francois-Xavier Briol
Author:
Mark Girolami
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