When Buffon meets Bertrand

When Peter Diggle gave his “short history” of spatial statistics this morning (I typed this in the taxi from Charles de Gaulle airport, after waiting one hour for my bag!), he started with a nice slide about Buffon’s needle (and Buffon’s portrait), since Julian Besag was often prone to give this problem as a final exam to Durham students (one of whom is responsible for the candidate’s formula). This started me thinking about how this was open to a Bertrand’s paradox of its own. Indeed, randomness for the needle throw can be represented in many ways:

• needle centre uniformly distributed over the room (or the perpendicular to the boards) with a random orientation (with a provision to have the needle fit);
• needle endpoint uniformly distributed over the room (again a uniform over the perpendicular is enough) with a random orientation (again with a constraint);
• random orientation from one corner of the room and a uniform location of the centre on the resulting line (with constraints on both ends for the needle to fit);
• random orientation from one corner of the room and a uniform location of one endpoint on the resulting line, plus a Bernoulli generation to decide on the orientation (with constraints on both ends for the needle to fit);
• &tc.

I did not have time to implement those different generation mechanisms in R, but have little doubt they should lead to different probabilities of intersection between the needle and one of the board separations. I actually found a web-page at the University of Alabama Huntsville addressing this problem through exercises (plus 20,000 related entries! Including von Mises‘ Probability, Statistics and Truth itself. A book I should read one of those days, following Andrew.). Note that each version corresponds to a physical mechanism. Thus that there is no way to distinguish between them. Had I time, I would also like to consider the limiting case when the room gets infinite as, presumably, some of those proposals would end up being identical.

A critical assessment now published

The review of Krzysztof Burdzy’s The Search for Certainty has been posted on Bayesian Analysis today. Given the submission date, this makes for an ultrafast acceptance and publishing! Thanks for the editorial team for this speedy processing. The additional comments by Andrew Gelman, Krzysztof Burdzy, and Larry Wasserman should follow soon. I am not sure I will be allowed for a reply to these comments, since they mostly state different opinions rather than facts about the book.

Comments on the reply

It is quite nice that Professor Burdzy replied to our critical analyses of his book The Search for Certainty in such a kind way, but, overall, my opinion about the book is basically unchanged, just as Andrew’s. The book does not bring much to the way I perceive Statistics nor to the reasons for which I favour Bayesian over frequentist analysis. So, just as clearly, I am not interested in (i) (philosophical) criticisms of von Mises and de Finetti philosophical bases, in (ii) a new “scientific theory of probability”, or in (iii) educating scientists about philosophical theories of probability. ’tis not to say that I cannot launch into a (pseudo-)philosophical defence of the Bayesian approach at times, but I then focus on the coherence of the Bayesian principles and on their fundamental  interaction with decision, taking for granted the measure-theoretic foundations of probability objects…

I think the other “Facts” in the reply are basically a matter of imprecision in my text or in Burdzy’s, which could be settled by a revision of my paper (and the choice of a less unlikely picture by the publisher!). The same applies to the “Opinion” part, as, again, I do not think The Search for Certainty has any bearing on statistical practice (and I agree with Professor Burdzy that most statisticians do not care much about those philosophical issues). I find the remark about catholic theology of interest in that it implies a coherent and unique (or dual) basis for the philosophical justification of probability. One most exciting aspect about statistics—and one that makes the field so distinct from mathematics—is on the opposite that the same problem can be tackled in many different ways, based on very different premises. There is no “Vatican doctrine”, thank God—easy pun, I know!

Anyway, I hope the sum of those discussions can eventually make it into a (BA?) discussion paper!

The Search for Certainty

“The expected value is hardly ever expected.” The Search for Certainty, page 207.

Following the advice of an Og’s reader, Erkan, I read Krzysztof Burdzy’s The Search for Certainty over the past weeks (and mostly in the métro). The author is primarily a probabilist, but he has set upon launching a radical criticism of the philosophical foundations of Statistics, the book subtitle being (somehow misleadingly) put as On the Clash of Science and Philosophy of Probability. Or so he thinks. (The introduction to the book is available on the author’s webpage and he has also started a blog related with the book. The cover is made of seven dices all showing sixes, with on top the probability p=0.0000036… of such an event, provided the dice is fair. And provided one does not take into account the suspicious fives all facing the reader!)

“Of the four well crystallized philosophies of probability, two chose the certainty as their intellectual holy grail. Those are the failed theories of von Mises and de Finetti.” The Search for Certainty, page 30.

Although I have read the book with a pen, I will not go at this stage into a detailed analysis of The Search for Certainty. Indeed, I found the book both annoying and unconvincing, for reasons not very different from the criticisms addressed at Taleb’s The Black Swan. The book aims at demonstrating that the philosophical arguments underlying both frequentist and Bayesian Statistics are wrong. Unsurprisingly (!), I find the book lacking in this demonstration and overall poor from a scholarly perspective. It compares with Taleb’s The Black Swan in that the attempts at philosophy are more related to everyday “common sense” than to deep (and scholarly) philosophy (and they also involve the apparently inevitable Karl Popper, “the champion of the propensity theory of probability”, p.43!). The main point made in The Search for Certainty is very narrow in that Burdzy concentrates on two very specific entries to frequentism and subjectivism, namely von Mises’ and de Finetti’s, respectively, while those are not your average statistician’s references. For instance, von Mises bases his definition of frequency properties on the notion of collectives, a notion I had not previously encountered. Similarly, de Finetti’s statement “Probability does not exist” cannot be seen as the core principle of many Bayesian statisticians and I certainly do not relate to his all-subjective perspective for conducting Bayesian inference.

“Mr. Winston is unique because we know something about him that we do not know about any other individual in the population.” The Search for Certainty, page 66.

The style of The Search for Certainty is fairly annoying, if of a different sort than Taleb‘s. It is predominantly non-technical, the worked-out examples being always of the coin and balls-in-an-urn type. Arguments are never more than one paragraph long and metaphors and weak analogies are more confusing than helpful. The book criticises a lot decision-theory and the related coherence of Bayesian procedures, ie against the Dutch Book argument, but it misses the connections between frequentist optimality(ies) and Bayesian procedures, like Wald’s complete class theorem, Welch and Peers (1964) matching priors, or the more recent Berger’s frequentist-Bayesian perspective. There are also (in my opinion) confusions, as when the basis for the frequency approach to probability is criticised for being connected with an unrealistic infinity of events, thus confusing concepts with experiments (indeed, the Large Law of Large Numbers cannot be proved by an experiment), the existence of a model versus its assessment (“Kolmogorov’s axioms say nothing on how to match the mathematical results with reality”, p.31), the use of a probability against its “truth”, the inclusion of time (and thus model shifts) into mathematical axioms, the assimilation of frequentist statistics to unbiased estimators, a somehow diffuse belief that some priors can be proven to be better than others, an argument that  they can be evaluated by their predictive performances… The insistence in adding new axioms to Kolmogorov’s is furthermore puzzling:

(L4) If there exists a symmetry on the space of possible outcomes which maps an event A onto an event B then the two events have equal probabilities, that is, P(A) = P(B).
(L5) An event has probability 0 if and only if it cannot occur. An event has probability 1 if and only if it must occur.

The axiom (L4) relates to both the Principle of Insufficient Reasons, whose limitations are the cause for much debate in the selection of prior distributions, and to invariance principles that lead to Haar measures as default noninformative priors. But I do not see the point in adding such an axiom into the tenets of probability. (And (L5) is more psychological than mathematical…)

“There is no justification for the use of the Bayes theorem in the subjective theory.” The Search for Certainty, page 144.

In conclusion, it is hopefully obvious that I did not overly enjoy this The Search for Certainty and that I do not consider it makes a significant contribution to the foundations of statistical inference and in particular to Bayesian analysis. Being examined by an outsider to our discipline certainly has a strong appeal, but only if done at a deep enough level.

Ps-Be warned that there is a homonymous book with full titlde The Search for Certainty: A Philosophical Account of Foundations of Mathematics on issues related with Gödel’s incompleteness theorem, by Marcus Gianquinto, book that I mistakenly bought for The Search for Certainty discussed here. Professor Gianquinto is actually a professor of philosophy at UCL, who specialises in epistemology so he could also comment on this book.