Semi-Exact Control Functionals From Sard’s Method

@article{South2020SemiExactCF,
  title={Semi-Exact Control Functionals From Sard’s Method},
  author={Leah South and Toni Karvonen and Christopher Nemeth and Mark A. Girolami and Chris J. Oates},
  journal={arXiv: Computation},
  year={2020}
}
This paper focuses on the numerical computation of posterior expected quantities of interest, where existing approaches based on ergodic averages are gated by the asymptotic variance of the integrand. To address this challenge, a novel variance reduction technique is proposed, based on Sard's approach to numerical integration and the control functional method. The use of Sard's approach ensures that our control functionals are exact on all polynomials up to a fixed degree in the Bernstein-von… Expand
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References

SHOWING 1-10 OF 84 REFERENCES
A Bayes-Sard Cubature Method
TLDR
Bayes-Sard cubature is introduced, a probabilistic framework that combines the flexibility of Bayesian cubature with the robustness of classical cubatures which are well-established, and two orders of magnitude reduction in error are reported in the context of a high-dimensional financial integral. Expand
Control functionals for Monte Carlo integration
Summary A non-parametric extension of control variates is presented. These leverage gradient information on the sampling density to achieve substantial variance reduction. It is not required thatExpand
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 aExpand
Scalable Control Variates for Monte Carlo Methods via Stochastic Optimization
TLDR
This paper considers control variates based on Stein operators, presenting a framework that encompasses and generalizes existing approaches that use polynomials, kernels and neural networks, leading to scalable and effective control Variates. Expand
Variance reduction for MCMC methods via martingale representations
In this paper we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete time martingale representation. Our approach is fullyExpand
A Riemann–Stein Kernel Method
This paper proposes and studies a numerical method for approximation of posterior expectations based on interpolation with a Stein reproducing kernel. Finite-sample-size bounds on the approximationExpand
Regularised Zero-Variance Control Variates for High-Dimensional Variance Reduction
Zero-variance control variates (ZV-CV) are a post-processing method to reduce the variance of Monte Carlo estimators of expectations using the derivatives of the log target. Once the derivatives areExpand
Bayes–Hermite quadrature
Abstract Bayesian quadrature treats the problem of numerical integration as one of statistical inference. A prior Gaussian process distribution is assumed for the integrand, observations arise fromExpand
The reproducing Stein kernel approach for post-hoc corrected sampling
Stein importance sampling is a widely applicable technique based on kernelized Stein discrepancy, which corrects the output of approximate sampling algorithms by reweighting the empiricalExpand
Diffusion approximations and control variates for MCMC
A new methodology is presented for the construction of control variates to reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC) samplers. Our control variates are defined asExpand
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