Non-Gaussian statistical inverse problems. Part I: Posterior distributions

Sari Lasanen

doi:10.3934/ipi.2012.6.215

Article Contents

2012, Volume 6, Issue 2: 215-266. Doi: 10.3934/ipi.2012.6.215

This issue Previous Article Surveillance video processing using compressive sensing Next Article Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns

Non-Gaussian statistical inverse problems. Part I: Posterior distributions

Sari Lasanen^1,

1.
Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu

Received: June 2009

Revised: January 2012

Published: May 2012

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Abstract

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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Mathematics Subject Classification: Primary: 60B10, 65J22; Secondary: 60B11, 62C10.

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