Improper versus finitely additive distributions as limits of countably additive probabilities

The Bayesian paradigm with proper priors can be extended either to improper distributions or to finitely additive probabilities (FAPs). Improper distributions and diffuse FAPs can be seen as limits of proper distribution sequences for specific convergence modes. In this paper, we compare these two kinds of limits. We show that improper distributions and FAPs represent two distinct features of the limit behavior of a sequence of proper distribution. More specifically, an improper distribution characterizes the behavior of the sequence inside the domain, whereas diffuse FAPs characterizes how the mass concentrates on the boundary of the domain. Therefore, a diffuse FAP cannot be seen as the counterpart of an improper distribution. As an illustration, we consider several approach to define uniform FAP distributions on natural numbers as an equivalent of improper flat prior. We also show that expected logarithmic convergence may depend on the chosen sequence of compact sets.

Automated summary

In this paper, the comparison between improper distributions and diffuse FAPs as limits of proper distribution sequences is discussed. Improper distributions characterize the behavior of the sequence inside the domain, while diffuse FAPs characterize how the mass concentrates on the boundary of the domain. Therefore, a diffuse FAP cannot be seen as the counterpart of an improper distribution.