A general framework for updating belief distributions

We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special ca...

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Vydané v:	Journal of the Royal Statistical Society. Series B, Statistical methodology Ročník 78; číslo 5; s. 1103 - 1130
Hlavní autori:	Bissiri, P. G., Holmes, C. C., Walker, S. G.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	England Blackwell Publishing Ltd 01.11.2016 John Wiley & Sons Ltd Oxford University Press John Wiley and Sons Inc
Predmet:	Bayesian analysis Bayesian theory Beliefs Coherence Data Decision theory Density equations Frame analysis General Bayesian updating Generalized estimating equations Gibbs posteriors Inference Information Joints Learning Loss function Mathematical models Maximum entropy Original Parameters probability Provably approximately correct Bayes methods Self-information loss function Statistics Studies Subjectivity Loss function Self‐information loss function Decision theory Provably approximately correct Bayes methods General Bayesian updating Gibbs posteriors Information Maximum entropy Generalized estimating equations
ISSN:	1369-7412, 1467-9868
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Popis
Shrnutí:	We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.
Bibliografia:	ArticleID:RSSB12158 'Supplementary material'. ark:/67375/WNG-V2WJ3ZB6-B istex:F8AE7E12093DC32F720E23FF17B2BFAFC61EC32E SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23
ISSN:	1369-7412 1467-9868
DOI:	10.1111/rssb.12158