Linear inverse problems for generalised random variables

Abstract

In a statistical inverse theory both the unknown quantity and the measurement are random variables. The solution of the inverse problem is then the conditional distribution of the unknown variable with the measurement supposed to be known. Both variables often have their values in spaces of functions or generalised functions while the statistical theory of conditional distributions has only been fully developed for Polish spaces. Also, the mappings representing the solution of linear inverse problems with Gaussian priors and noises are usually only defined on a subset of the spaces used. This problem has previously been correctly handled only for Hilbert-space-valued variables. The existence of a regular version of the conditional distribution of random variables with values in spaces of generalised functions is shown and the inverse problem is solved in the linear Gaussian case.