期刊
INVERSE PROBLEMS AND IMAGING
卷 6, 期 2, 页码 215-266出版社
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/ipi.2012.6.215
关键词
Statistical inverse problems; posterior distributions; Bayesian methods; measures on linear spaces; non-Gaussian distributions
资金
- Academy of Finland (Finnish Programme for Centres of Excellence in Research) [213476]
- Finnish Programme for Centres of Excellence in Research
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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