Analysis of the Gibbs Sampler for Hierarchical Inverse Problems
Analysis of the Gibbs Sampler for Hierarchical Inverse Problems
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DOI：
10.1137/130944229
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发表时间：
2014-01-01
影响因子：
2
通讯作者：
Stuart, Andrew M.
中科院分区：
文献类型：
--
作者：
Agapiou, Sergios;Bardsley, Johnathan M.;Stuart, Andrew M.
在实际应用中，许多反问题都是由未知参数为场的连续介质模型引起的。在实践中，未知字段被离散化，导致RN中的问题，理解为细化离散化，即增加N，通常是可取的。在贝叶斯反演的背景下，这种情况表明了两个问题的重要性：（i）以这样一种方式定义超参数，即它们在连续极限N ->无穷大中是可解释的，并且它们的值可以在不同的离散化水平之间进行比较;（ii）理解探测后验分布作为大N的函数的算法的效率。在这里，我们解决这两个问题的背景下，线性逆问题的加性高斯噪声在一个分层建模框架的基础上高斯先验的未知领域和反伽玛先验的超参数，即振幅的先验方差。该模型的结构是这样的吉布斯采样器可以很容易地实现探测后验分布。订阅的教条，一个人应该认为在有限维实现之前，无限维，我们提出了函数空间的直觉，并提供严格的理论表明，随着N的增加，吉布斯采样器的组件采样的振幅的先验方差变得越来越慢。我们讨论了一个重新参数化的先验方差，是强大的维数的增加，我们给出的数值实验表明，我们的重新参数化防止放缓。我们对先验超参数的行为的直觉，有和没有重新参数化，是足够普遍的，包括广泛的一类非线性反问题，以及其他家庭的超先验。
Many inverse problems arising in applications come from continuum models where the unknown parameter is a field. In practice the unknown field is discretized, resulting in a problem in RN, with an understanding that refining the discretization, that is, increasing N, will often be desirable. In the context of Bayesian inversion this situation suggests the importance of two issues: (i) defining hyperparameters in such a way that they are interpretable in the continuum limit N -> infinity and so that their values may be compared between different discretization levels; and (ii) understanding the efficiency of algorithms for probing the posterior distribution as a function of large N. Here we address these two issues in the context of linear inverse problems subject to additive Gaussian noise within a hierarchical modeling framework based on a Gaussian prior for the unknown field and an inverse-gamma prior for a hyperparameter, namely the amplitude of the prior variance. The structure of the model is such that the Gibbs sampler can be easily implemented for probing the posterior distribution. Subscribing to the dogma that one should think infinite-dimensionally before implementing in finite dimensions, we present function space intuition and provide rigorous theory showing that as N increases, the component of the Gibbs sampler for sampling the amplitude of the prior variance becomes increasingly slower. We discuss a reparametrization of the prior variance that is robust with respect to the increase in dimension; we give numerical experiments which exhibit that our reparametrization prevents the slowing down. Our intuition on the behavior of the prior hyperparameter, with and without reparametrization, is sufficiently general to include a broad class of nonlinear inverse problems as well as other families of hyperpriors.