Abstract
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 approximation error are established for posterior distributions supported on a compact Riemannian manifold, and we relate these to a kernel Stein discrepancy (KSD). Moreover, we prove in our setting that the KSD is equivalent to Sobolev discrepancy and, in doing so, we completely characterise the convergence-determining properties of KSD. Our contribution is rooted in a novel combination of Stein’s method, the theory of reproducing kernels, and existence and regularity results for partial differential equations on a Riemannian manifold.
Acknowledgements
CJO and MG were supported by the Lloyd’s Register Foundation programme on data-centric engineering at the Alan Turing Institute, UK. AB was supported by a Roth scholarship from the Department of Mathematics at Imperial College London, UK. EP was partially supported by FONDECYT Grant [1170290], Chile, and by Iniciativa Cientiífica Milenio - Minecon Nucleo Milenio MESCD. MG was supported by the EPSRC grants [EP/K034154/1, EP/R018413/1, EP/P020720/1, EP/L014165/1], an EPSRC Established Career Fellowship [EP/J016934/1] and a Royal Academy of Engineering Research Chair in Data Centric Engineering. The authors are grateful for discussions with Andrew Duncan, Toni Karvonen, Chang Liu, Gustav Holzegel, Julio Delgado and Andrew Stuart, and to an Associate Editor and Reviewer for constructive feedback on an earlier version of the manuscript.
Citation
Alessandro Barp. Chris. J. Oates. Emilio Porcu. Mark Girolami. "A Riemann–Stein kernel method." Bernoulli 28 (4) 2181 - 2208, November 2022. https://doi.org/10.3150/21-BEJ1415
