Conditioning as disintegration

TLDR

Conditional probability distributions are often treated with technical rigor using Radon–Nikodym derivatives and measurability arguments, which obscure intuitive understanding. The authors aim to provide a rigorous, guilt‑free framework for manipulating conditional distributions by treating them as disintegrating measures. They develop a theory and illustrate it with examples from EM algorithms, Neyman factorization, Bayes theory, and marginalization paradoxes.

Abstract

Conditional probability distributions seem to have a bad reputation when it comes to rigorous treatment of conditioning. Technical arguments are published as manipulations of Radon–Nikodym derivatives, although we all secretly perform heuristic calculations using elementary definitions of conditional probabilities. In print, measurability and averaging properties substitute for intuitive ideas about random variables behaving like constants given particular conditioning information. One way to engage in rigorous, guilt‐free manipulation of conditional distributions is to treat them as disintegrating measures—families of probability measures concentrating on the level sets of a conditioning statistic. In this paper we present a little theory and a range of examples—from EM algorithms and the Neyman factorization, through Bayes theory and marginalization paradoxes—to suggest that disintegrations have both intuitive appeal and the rigor needed for many problems in mathematical statistics.