Bayesian analysis in inverse problems

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Abstract
Considers some statistical aspects of inverse problems, using Bayesian analysis, particularly estimation and hypothesis-testing questions for parameter-dependent differential equations. The author relates Bayesian maximum likelihood to Tikhonov regularization and applies the expectation-minimization algorithm to the problem of setting regularization levels. Further, he compares 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. He develops a framework for infinite dimensional Bayesian analysis, including convergence of approximations required to perform inference tasks computationally.