# Preprint: Robust Bayesian Meta-Analysis: Addressing Publication Bias with Model-Averaging

This post is a teaser for Maier, Bartoš, & Wagenmakers (2020). Robust Bayesian meta-analysis: Addressing publication bias with model-averaging. Preprint available on PsyArXiv: https://psyarxiv.com/u4cns

### Abstract

“Meta-analysis is an important quantitative tool for cumulative science, but its application is frustrated by publication bias. In order to test and adjust for publication bias, we extend model-averaged Bayesian meta-analysis with selection models. The resulting Robust Bayesian Meta-analysis (RoBMA) methodology does not require all-or-none decisions about the presence of publication bias, is able to quantify evidence in favor of the absence of publication bias, and performs well under high heterogeneity. By model-averaging over a set of 12 models, RoBMA is relatively robust to model misspecification, and simulations show that it outperforms existing methods. We demonstrate that RoBMA finds evidence for the absence of publication bias in Registered Replication Reports and reliably avoids false positives. We provide an implementation in R and JASP so that researchers can easily apply the new methodology to their own data.”
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# Struggling with de Finetti’s Representation Theorem

De Finetti’s Representation Theorem is among the most celebrated results in Bayesian statistics. As I mentioned in an earlier post, I have never really understood its significance. A host of excellent writers have all tried to explain why the result is so important [e.g., Lindley (2006, pp. 107-109), Diaconis & Skyrms (2018, pp. 122-125), and the various works by Zabell], but their words just went over my head. Yes, I understand that for an exchangeable series, the probability of the data can be viewed as a weighted mixture over a prior distribution, but this just seemed like an application of Bayes rule — you integrate out the parameter to obtain the result. So what’s the big deal?

Recently I stumbled across a 2004 article by Phil Dawid, one of the most reputable (and original) Bayesian statisticians. In his article, Dawid provides a relatively accessible introduction to the importance of de Finetti’s theorem. In the section “Exchangeability”, Dawid writes:
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# PROBABILITY DOES NOT EXIST (Part V): De Finetti’s 1974 Preface (Part III)

This is the third and final post on the 1974 preface of Bruno de Finetti’s masterpiece “Theory of Probability”, which is missing from the reprint of the 1970 book. Below, the use of italics is always as in the original text.

### De Finetti’s Preface Continued [Annotated]

“It would be impossible, even if space permitted, to trace back the possible development of my ideas, and their relationships with more or less similar positions held by other authors, both past and present. A brief survey is better than nothing, however (even though there is an inevitable arbitrariness in the selection of names to be mentioned).
I am convinced that my basic ideas go back to the years of High School as a result of my preference for the British philosophers Locke, Berkeley and, above all, Hume! I do not know to what extent the Italian school textbooks and my own interpretations were valid: I believe that my work based on exchangeability corresponds to Hume’s ideas, but some other scholars do not agree. I was also favourably impressed, a few years later, by the ideas of Pragmatism, and the related notions of operational definitions in Physics. I particularly liked the Pragmatism of Giovanni Vailati—who somehow Italianized’ James and Peirce—and, as for operationalism, I was very much struck by Einstein’s relativity of simultaneity’, and by Mach and (later) Bridgman.
As far as Probability is concerned, the first book I encountered was that of Czuber. (Before 1950—my first visit to the USA—I did not know any English, but only German and French.) For two or three years (before and after the `Laurea’ in Mathematics, and some application of probability to research on Medelian heredity), I attempted to find valid foundations for all the theories mentioned, and I reached the conclusion that the classical and frequentist theories admitted no sensible foundation, whereas the subjectivistic one was fully justified on a normative-behaviouristic basis.”

# PROBABILITY DOES NOT EXIST (Part IV): De Finetti’s 1974 Preface (Part II)

Here we continue our coverage of the 1974 preface of Bruno de Finetti’s masterpiece “Theory of Probability”, which is missing from the reprint of the 1970 book. Multiple posts are required to cover the entire preface. Below, the use of italics is always as in the original text.

### De Finetti’s Preface, Continued [Annotated]

“The numerous, different, opposed attempts to put forward particular points of view which, in the opinion of their supporters, would endow Probability Theory with a ‘nobler’ status, or a ‘more scientific’ character, or ‘firmer’ philosophical or logical foundations, have only served to generate confusion and obscurity, and to provoke well-known polemics and disagreements–even between supporters of essentially the same framework.
The main points of view that have been put forward are as follows.
The classical view, based on physical considerations of symmetry, in which one should be obliged to give the same probability to such ‘symmetric’ cases. But which symmetry? And, in any case, why? The original sentence becomes meaningful if reversed: the symmetry is probabilistically significant, in someone’s opinion, if it leads him to assign the same probabilities to such events.”