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Analysis of the Gibbs Sampler for hierarchical inverse problems
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Agapiou, Sergios, Bardsley, Johnathan M., Papaspiliopoulos, Omiros and Stuart, A. M. (2014) Analysis of the Gibbs Sampler for hierarchical inverse problems. SIAM/ASA Journal on Uncertainty Quantification, 2 (1). pp. 511-544. doi:10.1137/130944229 ISSN 2166-2525.
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Official URL: http://dx.doi.org/10.1137/130944229
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.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Gaussian processes, Markov processes, Monte Carlo method, Uncertainty -- Mathematical models | ||||||||
| Journal or Publication Title: | SIAM/ASA Journal on Uncertainty Quantification | ||||||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||||||
| ISSN: | 2166-2525 | ||||||||
| Official Date: | 23 September 2014 | ||||||||
| Dates: |
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| Volume: | 2 | ||||||||
| Number: | 1 | ||||||||
| Page Range: | pp. 511-544 | ||||||||
| DOI: | 10.1137/130944229 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Date of first compliant deposit: | 3 March 2017 | ||||||||
| Date of first compliant Open Access: | 3 March 2017 |
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