Nonparametric Bayesian inference for reversible multi-dimensional diffusions
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We study nonparametric Bayesian modelling of reversible multi-dimensional diffusions with periodic drift. For continuous observations 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 standard Gaussian priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension. We further establish contraction rates for p-exponential priors over spatially inhomogeneous Besov function classes for which Gaussian priors are unsuitable, again in any dimension.
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