Abstract
We study nonparametric Bayesian models for reversible multidimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and p-exponential priors, which are shown to converge to the truth at the optimal nonparametric rate over Sobolev smoothness classes in any dimension.
Funding Statement
M.G. was supported by the European Research Council under ERC grant agreement No. 647812 (UQMSI), and during part of the revision of the manuscript was affiliated with the University of Oxford and supported by the ERC grant agreement No. 834275 (GTBB).
Acknowledgements
We would like to thank Andrew Stuart for raising the question that led to this research, Richard Nickl for valuable discussions, and the Associate Editor and three referees for many helpful comments that improved the manuscript.
Citation
Matteo Giordano. Kolyan Ray. "Nonparametric Bayesian inference for reversible multidimensional diffusions." Ann. Statist. 50 (5) 2872 - 2898, October 2022. https://doi.org/10.1214/22-AOS2213
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