Inverse problems as statistics

Abstract

What mathematicians, scientists, engineers and statisticians mean by ‘inverseproblem’ differs. For a statistician, an inverse problem is an inference orestimation problem. The data are finite in number and contain errors, as they doin classical estimation or inference problems, and the unknown typically is infinitedimensional, as it is in nonparametric regression. The additional complication inan inverse problem is that the data are only indirectly related to the unknown.Canonical abstract formulations of statistical estimation problems subsume thiscomplication by allowing probability distributions to be indexed in more-or-lessarbitrary ways by parameters, which can be infinite dimensional. Standardstatistical concepts, questions and considerations such as bias, variance,mean-squared error, identifiability, consistency, efficiency and variousforms of optimality apply to inverse problems. This paper discusses inverseproblems as statistical estimation and inference problems, and points to theliterature for a variety of techniques and results. It shows how statisticalmeasures of performance apply to techniques used in practical inverseproblems, such as regularization, maximum penalized likelihood, Bayesestimation and the Backus–Gilbert method. The paper generalizes results ofBackus and Gilbert characterizing parameters in inverse problems thatcan be estimated with finite bias. It also establishes general conditionsunder which parameters in inverse problems can be estimated consistently.