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British Journal of Mathematical and Statistical Psychology (2012)

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2012 The British Psychological Society 

TheBritishPsychologicalSociety

www.wileyonlinelibrary.com

Philosophy and the practice of Bayesian statistics

Andrew Gelman

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and Cosma Rohilla Shalizi

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Department of Statistics and Department of Political Science, Columbia University,New York, USA

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Statistics Department, Carnegie Mellon University, Santa Fe Institute, Pittsburgh,USA

A substantial school in the philosophy of science identifies Bayesian inference withinductive inference and even rationality as such, and seems to be strengthened by therise and practical success of Bayesian statistics. We argue that the most successfulforms of Bayesian statistics do not actually support that particular philosophy but ratheraccord much better with sophisticated forms of hypothetico-deductivism. We examinethe actual role played by prior distributions in Bayesian models, and the crucial aspects of modelcheckingandmodelrevision,whichfalloutsidethescopeofBayesianconfirmationtheory. We draw on the literature on the consistency of Bayesian updating and also onour experience of applied work in social science. Clarity about these matters shouldbenefit not just philosophy of science, but also statistical practice. At best, the inductivistview has encouraged researchers to fit and compare models without checking them; atworst,theoristshaveactivelydiscouragedpractitionersfromperformingmodelcheckingbecause it does not fit into their framework.

1. The usual story – which we don’t like

In so far as I have a coherent philosophy of statistics, I hope it is ‘robust’ enough to copein principle with the whole of statistics, and sufficiently undogmatic not to imply that allthose who may think rather differently from me are necessarily stupid. If at times I do seemdogmatic, it is because it is convenient to give my own views as unequivocally as possible.(Bartlett, 1967, p. 458)

Schools of statistical inference are sometimes linked to approaches to the philosophy of science. ‘Classical’ statistics – as exemplified by Fisher’s

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-values, Neyman–Pearsonhypothesistests,andNeyman’sconfidenceintervals–isassociatedwiththehypothetico-deductive and falsificationist view of science. Scientists devise hypotheses, deduce im-plications for observations from them, and test those implications. Scientific hypotheses

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Correspondence should be addressed to Andrew Gelman, Department of Statistics and Departmentof Political Science, 1016 Social Work Bldg, Columbia University, New York, NY 10027 USA (e-mail:gelman@stat.columbia.edu).

DOI:10.1111/j.2044-8317.2011.02037.x