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:我们用角中高斯前科对高斯星界进行高方位分析。 这些前科模型的反对称方向数据很容易在希尔伯特空间进行定义,并发生于巴伊西亚二进制分类和水平定置倒置中。 在本文中,我们得出了Markov链的高效方法,用以对这些前科进行近似次子采样。 我们的方法依赖于将取样问题移到环绕的希尔伯特空间,并利用线性空间中现有的维度独立的采样器。 通过推向方向的马尔科夫内核构造,我们随后在球体上获得了马尔科夫链,这继承了线性空间取样员的可逆性和光谱差距特性。 此外,我们提议的算法在数字实验中显示了维度独立的效率。