BAYESIAN-ANALYSIS IN INVERSE PROBLEMS

In this paper, we consider some statistical aspects of inverse problems, using Bayesian analysis, particularly estimation and hypothesis-testing questions for parameter-dependent differential equations. We relate Bayesian maximum likelihood to Tikhonov regularization, and we apply the expectation-minimization (E-M) algorithm to the problem of setting regularization levels. Further, we compare Bayesian results with those of a classical statistical approach, through consistency and asymptotic normality. A numerical example illustrates the application of Bayesian techniques. In many cases one is interested in parameters which are infinite dimensional (e.g. functions). Bayesian techniques offer a sound theoretical and computational paradigm, through probability measures on Banach space. We develop a framework for infinite dimensional Bayesian analysis, including convergence of approximations required to perform inference tasks computationally.