Neerlandica Conditioning as disintegrationbyJoseph
نویسندگان
چکیده
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 deenitions 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 make a case for disintegration as a desirable part of the education of every graduate student of probability and statistics. We present a little theory and a range of examples|from EM algorithms and the Neyman factorization, through Bayes theory and marginalization paradoxes, to martin-gales adapted to Brownian ltrations|to suggest that disintegrations have both intuitive appeal and the rigor needed for many problems in mathematical statistics.
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