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 density estimation and binary 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.

中文翻译：

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球面上的维度无关马尔可夫链蒙特卡罗

我们考虑对具有角度中心高斯先验的高维球体进行贝叶斯分析。这些先验模型对映对称方向数据很容易在希尔伯特空间中定义，并且发生在例如贝叶斯密度估计和二进制水平集反演中。在本文中，我们推导了有效的马尔可夫链蒙特卡罗方法，用于根据这些先验对后验进行近似采样。我们的方法依赖于将采样问题提升到环境希尔伯特空间，并利用线性空间中现有的与维度无关的采样器。通过前推马尔可夫核构造，我们获得了球体上的马尔可夫链，它继承了线性空间中采样器的可逆性和谱间隙属性。此外，我们提出的算法在数值实验中显示出与维度无关的效率。