NON-GAUSSIAN STATISTICAL INVERSE PROBLEMS. PART I: POSTERIOR DISTRIBUTIONS

作者：Lasanen Sari^*

来源：Inverse Problems and Imaging, 2012, 6(2): 215-266.

DOI：10.3934/ipi.2012.6.215

• Statistical inverse problems
• posterior distributions
• Bayesian methods
• measures on linear spaces
• non-Gaussian distributions

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. %26lt;br%26gt;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. %26lt;br%26gt;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.

• 出版日期2012-5