We consider nonparametric Bayesian inference in a multidimensional diffusion model with reflecting boundary conditions based on discrete high-frequency observations. We prove a general posterior contraction rate theorem in $L^2$-loss, which is applied to Gaussian priors. The resulting posteriors, as well as their posterior means, are shown to converge to the ground truth at the minimax optimal rate over H\"older smoothness classes in any dimension. Of independent interest and as part of our proofs, we show that certain frequentist penalized least squares estimators are also minimax optimal.

翻译：我们考虑基于离散高频观测数据、具有反射边界条件的多维扩散模型中的非参数贝叶斯推断。我们证明了$L^2$损失下的一般后验收缩率定理，并将其应用于高斯先验。结果表明，所得后验分布及其后验均值在任意维度的H\"older光滑类上均以极小极大最优速率收敛于真实参数。作为证明的组成部分且具有独立意义的是，我们还证明了某些频率学派惩罚最小二乘估计量同样具有极小极大最优性。