Improper Priors via Expectation Measures

In Bayesian statistics, the prior distributions play a key role in the inference, and there are procedures for finding prior distributions. An important problem is that these procedures often lead to improper prior distributions that cannot be normalized to probability measures. Such improper prior distributions lead to technical problems, in that certain calculations are only fully justified in the literature for probability measures or perhaps for finite measures. Recently, expectation measures were introduced as an alternative to probability measures as a foundation for a theory of uncertainty. Using expectation theory and point processes, it is possible to give a probabilistic interpretation of an improper prior distribution. This will provide us with a rigid formalism for calculating posterior distributions in cases where the prior distributions are not proper without relying on approximation arguments.