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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Bibliographic Details
Published in:Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 78; no. 5; pp. 1103 - 1130
Main Authors: Bissiri, P. G., Holmes, C. C., Walker, S. G.
Format: Journal Article
Language:English
Published: England Blackwell Publishing Ltd 01.11.2016
John Wiley & Sons Ltd
Oxford University Press
John Wiley and Sons Inc
Subjects:
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
Online Access:Get full text
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Summary: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.
Bibliography:ArticleID:RSSB12158
'Supplementary material'.
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ISSN:1369-7412
1467-9868
DOI:10.1111/rssb.12158