Analysis of the Gibbs Sampler for Hierarchical Inverse Problems

TLDR

Inverse problems frequently arise from continuum models where the unknown field is discretized, and refining the discretization (increasing N) is desirable, raising issues of hyperparameter interpretation and algorithm efficiency in the Bayesian setting. The study aims to address hyperparameter interpretation and algorithm efficiency for linear inverse problems with Gaussian noise using a hierarchical Gaussian prior and inverse‑gamma hyperprior. The authors employ a hierarchical Gaussian–inverse‑gamma model that admits an easily implemented Gibbs sampler, analyze its behavior as N grows, and show that sampling the prior variance slows down but can be mitigated by a dimension‑robust reparametrization supported by theory and numerical experiments. The reparametrization of the prior variance prevents the Gibbs sampler’s slowdown as N grows, and the authors’ insights generalize to nonlinear inverse problems and other hyperprior families.

Abstract

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 $\mathbb{R}^N$, 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 $N \to \infty$ 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.