Title: Infinitesimal Distributions, Improper Priors and Bayesian Inference

Author(s): Leendert Huisman
Issue: Volume 78 Series A Part 2 Year 2016
Pages: 324 -- 346
In this article, I develop a generalization of probability distributions that is rich enough to include distributions that can act as suitable replacements for locally integrable improper distributions. These replacements have unit weight but share with the improper distributions from which they originate the essential behavioral properties for which the latter were constructed. Their theory will be developed and associated posterior (generalized) distributions will be obtained. These posterior probability distributions are close to those obtained from the improper ones by a formal application of Bayes’ rule; they differ, in fact, by only an infinitesimal correction. This correction is small enough that Bayesian parametric inference can be performed using the formal posteriors without having to consider the possibility of strong inconsistency or marginalization paradoxes. For more general tasks involving the posteriors, the correction may have to be taken into account.
AMS (2000) subject classification. Primary 62F15; Secondary 60E05.
Keywords and phrases: Bayesian inference, Improper priors, Marginalization paradox, Strong inconsistency.