Non-Gaussian statistical inverse problems. Part I: Posterior distributions (Q435839)

Given a probability space \((\Omega,\Sigma,P)\), the unknown \(X\) and its observation \(Y\) are modelled as random variables taking values in locally convex Souslin topological vector spaces \(F\) and \(G\), respectively. The observations are of the form \(Y=L(X)+\varepsilon\), where \(X\) and the random noise \(\varepsilon\) are independent and \(L:F\to G\) is continuous. The author provides the following definition of an inverse problem, which is well suited for a Bayesian framework: The statistical inverse problem is to estimate the probability distribution of the unknown given its randomly perturbed indirect observation.NEWLINENEWLINEThe present paper deals with following two topics: (i) Applicability of the generalized Bayes formula for statistical inverse problems in Souslin topological vector spaces, and (ii) well--posedness of the Bayesian statistical inverse problem.NEWLINENEWLINE While the original problem may be infinite-dimensional, Bayesian computatios are usually carried out in finite-dimensional cases. The author investigates the question of the well-posedness of the infinite-dimensional statistical inverse problem and shows that the continuous dependence of the posterior probabilities on the realizations of the observations provides a certain degree of uniqueness for the posterior distribution. Special emphasis is put on non-Gaussian noise.NEWLINENEWLINE An idea of the contents of this rich paper may be given by listing the section headers. After the Introduction the sections are: 2. Conditional probabilities and posterior distributions. 3. The representation of posterior distributions. 4. Examples of noise. Finally, it must be stressed that (the 10 page long) section 1.3 is devoted to an accurate review of the relevant literature and to the history of the problem.