Journal
INVERSE PROBLEMS AND IMAGING
Volume 6, Issue 2, Pages 215-266Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/ipi.2012.6.215
Keywords
Statistical inverse problems; posterior distributions; Bayesian methods; measures on linear spaces; non-Gaussian distributions
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Funding
- Academy of Finland (Finnish Programme for Centres of Excellence in Research) [213476]
- Finnish Programme for Centres of Excellence in Research
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One approach to noisy inverse problems is to use Bayesian methods. In this work, the statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect infinite-dimensional observation is studied in the Bayesian framework. The motivation for the work arises from the fact that the Bayesian computations are usually carried out in finite-dimensional cases, while the original inverse problem is often infinite-dimensional. A good understanding of an infinite-dimensional problem is, in general, helpful in finding efficient computational approaches to the problem. The fundamental question of well-posedness of the infinite-dimensional statistical inverse problem is considered. In particular, it is shown that the continuous dependence of the posterior probabilities on the realizations of the observation provides a certain degree of uniqueness for the posterior distribution. Special emphasis is on finding tools for working with non-Gaussian noise models. Especially, the applicability of the generalized Bayes formula is studied. Several examples of explicit posterior distributions are provided.
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