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Conditioning as disintegration

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  • J. T. Chang
  • D. Pollard

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.

Suggested Citation

  • J. T. Chang & D. Pollard, 1997. "Conditioning as disintegration," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 51(3), pages 287-317, November.
    • Handle: RePEc:bla:stanee:v:51:y:1997:i:3:p:287-317
    DOI: 10.1111/1467-9574.00056
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    Cited by:

    1. Marcel Klatt & Axel Munk & Yoav Zemel, 2022. "Limit laws for empirical optimal solutions in random linear programs," Annals of Operations Research, Springer, vol. 315(1), pages 251-278, August.
    2. Georg Nöldeke & Larry Samuelson, 2018. "The Implementation Duality," Econometrica, Econometric Society, vol. 86(4), pages 1283-1324, July.
    3. Gunsilius, Florian F., 2023. "A condition for the identification of multivariate models with binary instruments," Journal of Econometrics, Elsevier, vol. 235(1), pages 220-238.
    4. Benjamin Christoffersen & David Lando & Søren Feodor Nielsen, 2022. "Estimating volatility in the Merton model: The KMV estimate is not maximum likelihood," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 1214-1230, October.
    5. D M Farewell & R M Daniel & S R Seaman, 2022. "Missing at random: a stochastic process perspective [Contribution to the discussion of ‘Longitudinal data with dropout: Objectives, assumptions and a proposal’ by P. J. Diggle, D. Farewell and R. H," Biometrika, Biometrika Trust, vol. 109(1), pages 227-241.
    6. Heifetz, Aviad, 2006. "The positive foundation of the common prior assumption," Games and Economic Behavior, Elsevier, vol. 56(1), pages 105-120, July.
    7. Brendan K. Beare, 2023. "Optimal measure preserving derivatives revisited," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 370-388, April.
    8. Beare, Brendan K. & Seo, Juwon, 2014. "Time Irreversible Copula-Based Markov Models," Econometric Theory, Cambridge University Press, vol. 30(5), pages 923-960, October.
    9. Atar, Rami & Shadmi, Yonatan, 2023. "Fluid limits for earliest-deadline-first networks," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 279-307.
    10. Jupp, P.E. & Regoli, G. & Azzalini, A., 2016. "A general setting for symmetric distributions and their relationship to general distributions," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 107-119.
    11. Burkhart, Michael C., 2019. "A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding," Thesis Commons 4j3fu, Center for Open Science.
    12. João Correia-da-Silva, 2008. "Agreeing to disagree in a countable space of equiprobable states," FEP Working Papers 260, Universidade do Porto, Faculdade de Economia do Porto.
    13. Eduardo A. Souza-Rodrigues, 2016. "Nonparametric Regression with Common Shocks," Econometrics, MDPI, vol. 4(3), pages 1-17, September.
    14. Soham R. Phade & Venkat Anantharam, 2023. "Learning in Games with Cumulative Prospect Theoretic Preferences," Dynamic Games and Applications, Springer, vol. 13(1), pages 265-306, March.
    15. Julian Sester, 2023. "On intermediate Marginals in Martingale Optimal Transportation," Papers 2307.09710, arXiv.org, revised Nov 2023.
    16. João Correia-da-Silva, 2010. "Agreeing to disagree in a countable space of equiprobable states of nature," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 291-302, October.
    17. Nathan Canen & Kyungchul Song, 2020. "A Decomposition Approach to Counterfactual Analysis in Game-Theoretic Models," Papers 2010.08868, arXiv.org, revised Jul 2024.

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