Measurements and infinite-dimensional statistical inverse theory

The most important ingredient of the statistical inverse theory is the indirect and noisy measurement of the unknown. Without the measurement, the formula for the posterior distribution becomes useless. However, inserting the measurement into the posterior distribution is not always simple. In the general setting, the posterior distribution is defined as a regular conditional probability. Hence it is known up to almost all measurements, which is inconvenient when we are given a single measurement. This shortage is covered in the finite-dimensional statistical inverse theory by fixing versions of probability density functions. A usual choice is to consider continuous probability density functions. Unfortunately, infinite-dimensional probability measures lack density functions which prohibits us from using the same method in the general setting. In this work, other possibilities for fixing the posterior distributions are discussed in the Gaussian framework. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)