Improper vs finitely additive distributions as limits of countably additive probabilities

TL;DR

This paper compares improper distributions and finitely additive probabilities in Bayesian statistics, demonstrating they are fundamentally different limits of proper distributions and cannot be directly connected through sequence convergence.

Contribution

It establishes that improper distributions and FAPs are distinct limit concepts, with improper distributions describing internal behavior and FAPs focusing on boundary mass concentration.

Findings

Improper distributions and FAPs cannot be connected via proper distribution sequences.

Sequences can be constructed to have the same FAP limits but different improper distribution limits.

Illustrates the challenge of defining a uniform FAP on natural numbers.

Abstract

In Bayesian statistics, improper distributions and finitely additive probabilities (FAPs) are the two main alternatives to proper distributions, i.e. countably additive probabilities. Both of them can be seen as limits of proper distribution sequences w.r.t. to some specific convergence modes. Therefore, some authors attempt to link these two notions by this means, partly using heuristic arguments. The aim of the paper is to compare these two kinds of limits. We show that improper distributions and FAPs represent two distinct characteristics of a sequence of proper distributions and therefore, surprisingly, cannot be connected by the mean of proper distribution sequences. More specifically, for a sequence of proper distribution which converge to both an improper distribution and a set of FAPs, we show that another sequence of proper distributions can be constructed having the same FAP limits and converging to any given improper distribution. This result can be mainly explained by the fact that improper distributions describe the behavior of the sequence inside the domain after rescaling, whereas FAP limits describe how the mass concentrates on the boundary of the domain. We illustrate our results with several examples and we show the difficulty to define properly a uniform FAP distribution on the natural numbers as an equivalent of the improper flat prior. MSC 2010 subject classifications: Primary 62F15; secondary 62E17,60B10.