We develop a fully intrinsic Bayesian framework for nonparametric regression on the unit sphere based on isotropic Gaussian field priors and the harmonic structure induced by the Laplace-Beltrami operator. Under uniform random design, the regression model admits an exact diagonalization in the spherical harmonic basis, yielding a Gaussian sequence representation with frequency-dependent multiplicities. Exploiting this structure, we derive closed-form posterior distributions, optimal spectral truncation schemes, and sharp posterior contraction rates under integrated squared loss. For Gaussian priors with polynomially decaying angular power spectra, including spherical Matérn priors, we establish posterior contraction rates over Sobolev classes, which are minimax-optimal under correct prior calibration. We further show that the posterior mean admits an exact variational characterization as a geometrically intrinsic penalized least-squares estimator, equivalent to a Laplace-Beltrami smoothing spline.

翻译：我们基于各向同性高斯场先验和由拉普拉斯-贝尔特拉米算子诱导的调和结构，为球面上的非参数回归建立了一个完全内蕴的贝叶斯框架。在均匀随机设计下，该回归模型允许在球谐基上进行精确对角化，从而得到一个具有频率相关多重性的高斯序列表示。利用这一结构，我们推导出闭式后验分布、最优谱截断方案以及在积分平方损失下的尖锐后验收缩率。对于具有多项式衰减角功率谱的高斯先验（包括球面Matérn先验），我们在Sobolev类上建立了后验收缩率，该收缩率在先验校准正确时达到极小极大最优。我们进一步证明，后验均值可作为几何内蕴的惩罚最小二乘估计量获得精确的变分表征，其等价于一个拉普拉斯-贝尔特拉米平滑样条。