Inverse problems as statistics

What mathematicians, scientists, engineers and statisticians mean by 'inverse problem' differs. For a statistician, an inverse problem is an inference or estimation problem. The data are finite in number and contain errors, as they do in classical estimation or inference problems, and the unknown typically is infinite dimensional, as it is in nonparametric regression. The additional complication in an inverse problem is that the data are only indirectly related to the unknown. Canonical abstract formulations of statistical estimation problems subsume this complication by allowing probability distributions to be indexed in more-or-less arbitrary ways by parameters, which can be infinite dimensional. Standard statistical concepts, questions and considerations such as bias, variance, mean-squared error, identifiability, consistency, efficiency and various forms of optimality apply to inverse problems. This paper discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of techniques and results. It shows how statistical measures of performance apply to techniques used in practical inverse problems, such as regularization, maximum penalized likelihood, Bayes estimation and the Backus–Gilbert method. The paper generalizes results of Backus and Gilbert characterizing parameters in inverse problems that can be estimated with finite bias. It also establishes general conditions under which parameters in inverse problems can be estimated consistently.