论文标题
歧管提升：将MCMC缩放到消失的噪声状态
Manifold lifting: scaling MCMC to the vanishing noise regime
论文作者
论文摘要
标准的马尔可夫链蒙特卡洛方法难以探索集中在低维结构附近的分布。这些病理自然发生在许多情况下。例如，当观察数据提供了高度信息时，或者当感兴趣的统计参数不可识别时，它们对于贝叶斯反问题建模和贝叶斯神经网络通常是常见的。在本文中，我们提出了一种将原始抽样问题转化为探索嵌入在较高维空间中的歧管上的分布的策略；与原始的后部相反，这种提升的分布在消失的噪声限制中保持弥散。我们采用受约束的哈密顿蒙特卡洛法，利用了该提升分布的流形几何形状，以执行有效的近似推断。我们在几个数值实验中证明，与竞争方法相反，我们提出的方法的采样效率不会退化，因为要探索的目标分布在低维结构附近。
Standard Markov chain Monte Carlo methods struggle to explore distributions that are concentrated in the neighbourhood of low-dimensional structures. These pathologies naturally occur in a number of situations. For example, they are common to Bayesian inverse problem modelling and Bayesian neural networks, when observational data are highly informative, or when a subset of the statistical parameters of interest are non-identifiable. In this paper, we propose a strategy that transforms the original sampling problem into the task of exploring a distribution supported on a manifold embedded in a higher dimensional space; in contrast to the original posterior this lifted distribution remains diffuse in the vanishing noise limit. We employ a constrained Hamiltonian Monte Carlo method which exploits the manifold geometry of this lifted distribution, to perform efficient approximate inference. We demonstrate in several numerical experiments that, contrarily to competing approaches, the sampling efficiency of our proposed methodology does not degenerate as the target distribution to be explored concentrates near low dimensional structures.
