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Conditioning as disintegration
Author(s) -
Chang J. T.,
Pollard D.
Publication year - 1997
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/1467-9574.00056
Subject(s) - conditioning , statistic , conditional probability , bayes' theorem , mathematics , range (aeronautics) , sufficient statistic , heuristic , mathematical economics , probability and statistics , econometrics , appeal , conditional probability distribution , computer science , bayesian probability , statistics , artificial intelligence , materials science , composite material , political science , law
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.