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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.”

# PROBABILITY DOES NOT EXIST (Part III): De Finetti’s 1974 Preface (Part I)

In an earlier blogpost I complained that the reprint of Bruno de Finetti’s masterpiece “Theory of Probability” concerns the 1970 version, and that the famous preface to the 1974 edition is missing. This blogpost provides an annotated version of this preface (de Finetti, 1974, pp. x-xiv). As the preface spans about four pages, it will take several posts to cover it all. Below, the use of italics is always as in the original text.

### De Finetti’s Preface [Annotated]

“Is it possible that in just a few lines I can achieve what I failed to achieve in my many books and articles? Surely not. Nevertheless, this preface affords me the opportunity, and I shall make the attempt. It may be that misunderstandings which persist in the face of refutations dispersed or scattered over some hundreds of pages can be resolved once and for all if all the arguments are pre-emptively piled up against them.”

# Book Review of “Bayesian Statistics the Fun Way”

The subtitle says it all: “Understanding statistics and probability with Star Wars, Lego, and rubber ducks”. And the author, Will Kurt, does not disappoint: the writing is no-nonsense, the content is understandable, the examples are engaging, and the Bayesian concepts are explained clearly. Here are some of the book’s features that I particularly enjoyed:
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