We consider Bayesian analysis on high-dimensional spheres with angular central Gaussian priors. These priors model antipodally symmetric directional data, are easily defined in Hilbert spaces and occur, for instance, in Bayesian binary classification and level set inversion. In this paper we derive efficient Markov chain Monte Carlo methods for approximate sampling of posteriors with respect to these priors. Our approaches rely on lifting the sampling problem to the ambient Hilbert space and exploit existing dimension-independent samplers in linear spaces. By a push-forward Markov kernel construction we then obtain Markov chains on the sphere, which inherit reversibility and spectral gap properties from samplers in linear spaces. Moreover, our proposed algorithms show dimension-independent efficiency in numerical experiments.

翻译：我们研究了具有角中心高斯先验的高维球面上的贝叶斯分析。此类先验可建模反对称极性方向数据，易于在希尔伯特空间中定义，并出现在贝叶斯二元分类和水平集反演等场景中。本文推导了针对这些先验的后验近似采样的高效马尔可夫链蒙特卡洛方法。我们的方法通过将采样问题提升至环境希尔伯特空间，并利用线性空间中现有的维度无关采样器。通过前推马尔可夫核构造，我们在球面上获得了马尔可夫链，这些链继承了线性空间采样器的可逆性和谱间隙性质。此外，所提出的算法在数值实验中显示出与维度无关的计算效率。