Conditional probability and improper priors

TL;DR

This paper develops a mathematical foundation for Bayesian statistics with improper priors, showing that improper laws can be incorporated into axioms and that different formulations are mathematically equivalent, impacting how inconsistencies are understood.

Contribution

It introduces a new axiomatic framework for improper priors using conditional probability spaces, clarifying their interpretation and resolving paradoxes in statistical foundations.

Findings

Improper laws can be included in axioms with modified calculation rules.

Theories based on improper laws and conditional probability spaces are mathematically equivalent.

The choice of formulation is a matter of personal preference, affecting interpretation of improper priors.

Abstract

The purpose of this paper is to present a mathematical theory that can be used as a foundation for statistics that include improper priors. This theory includes improper laws in the initial axioms and has in particular Bayes theorem as a consequence. Another consequence is that some of the usual calculation rules are modified. This is important in relation to common statistical practice which usually include improper priors, but tends to use unaltered calculation rules. In some cases the results are valid, but in other cases inconsistencies may appear. The famous marginalization paradoxes exemplify this latter case. An alternative mathematical theory for the foundations of statistics can be formulated in terms of conditional probability spaces. In this case the appearance of improper laws is a consequence of the theory. It is proved here that the resulting mathematical structures for the two theories are equivalent. The conclusion is that the choice of the first or the second formulation for the initial axioms can be considered a matter of personal preference. Readers that initially have concerns regarding improper priors can possibly be more open toward a formulation of the initial axioms in terms of conditional probabilities. The interpretation of an improper law is given by the corresponding conditional probabilities. Keywords: Axioms of statistics, Conditional probability space, Improper prior, Projective space