17 equations that changed the World (#2)

(continuation of the book review)

“If you placed your finger at that point, the two halves of the string would still be able to vibrate in the sin 2x pattern, but not in the sin x one. This explains the Pythagorean discovery that a string half as long produced a note one octave higher.” (p.143)

The following chapters are all about Physics: the wave equation, Fourier’s transform and the heat equation, Navier-Stokes’ equation(s), Maxwell’s equation(s)—as in  The universe in zero word—, the second law of thermodynamics, E=mc² (of course!), and Schrödinger’s equation. I won’t go so much into details for those chapters, even though they are remarkably written. For instance, the chapter on waves made me understand the notion of harmonics in a much more intuitive and lasting way than previous readings. (This chapter 8 also mentions the “English mathematician Harold Jeffreys“, while Jeffreys was primarily a geophysicist. And a Bayesian statistician with major impact on the field, his Theory of Probability arguably being the first modern Bayesian book. Interestingly, Jeffreys also was the first one to find approximations to the Schrödinger’s equation, however he is not mentioned in this later chapter.) Chapter 9 mentions the heat equation but is truly about Fourier’s transform which he uses as a tool and later became a universal technique. It also covers Lebesgue’s integration theory, wavelets, and JPEG compression. Chapter 10 on Navier-Stokes’ equation also mentions climate sciences, where it takes a (reasonable) stand. Chapter 11 on Maxwell’s equations is a short introduction to electromagnetism, with radio the obvious illustration. (Maybe not the best chapter in the book.)

not only defended but also applied [to appear]

Our paper with Andrew Gelman, “Not only defended but also applied”: the perceived absurdity of Bayesian inference, has been reviewed for the second time and is to appear in The American Statistician, as a discussion paper. Terrific news! This is my first discussion paper in The American Statistician (and the second in total, the first one being the re-read of Jeffreys‘ Theory of Probability.) [The updated version is now on arXiv.]

PLoS topic page on ABC

A few more comments on the specific entry on ABC written by Mikael Sunnåker et al…. The entry starts with the representation of the posterior probability of an hypothesis, rather than with the posterior density of a model parameter, which seems to lead the novice reader astray. After all, (a) ABC was not introduced for conducting model choice and (b) interchanging hypothesis and model means that the probability of an hypothesis H as used in the entry is actually the evidence in favour of the corresponding model. (There are a few typos and grammar mistakes, but I assume either PLoS or later contributors will correct those.) When the authors state that the “outcome of the ABC rejection algorithm is a set of parameter estimates distributed according to the desired posterior distribution”, I think they are misleading the readers as they forget the “approximative” aspect of this distribution. Further below, I would have used the title “Insufficient summary statistics” rather than “Sufficient summary statistics”, as it spells out more clearly the fundamental issue with the potential difficulty in using ABC. (And I am not sure the subsequent paragraph on “Choice and sufficiency of summary statistics” should bother with the sufficiency aspects… It seems to me much more relevant to assess the impact on predictive performances.)

Although this is most minor, I would not have made mention of the (rather artificial) “table for interpretation of the strength in values of the Bayes factor (…) originally published by Harold Jeffreys^[6] “. I obviously appreciate very much that the authors advertise our warning about the potential lack of validity of an ABC based Bayes factor! I also like the notion of “quality control”, even though it should only appear once. And the pseudo-example is quite fine as an introduction, while it could be supplemented with the outcome resulting from a large n, to be compared with the true posterior distribution. The section “Pitfalls and remedies” is remarkable in that it details the necessary steps for validating a ABC implementation: the only entry I would remove is the one about “Prior distribution and parameter ranges”, in that this is not a problem inherent to ABC… (Granted, the authors present this as a “general risks in statistical inference exacerbated in ABC”, which makes more sense!) It may be that the section on the non-zero tolerance should emphasize more clearly the fact that ε should not be zero. As discussed in the recent Read Paper by Fearnhead and Prangle when envisioning ABC as a non-parametric method of inference.

At last, it is always possible to criticise the coverage of the historical part, since ABC is such a recent field that it is constantly evolving. But the authors correctly point out to (Don) Rubin on the one hand and to Diggle and Graton on the other. Now, I would suggest adding in this section links to the relevant softwares like our own DIY-ABC…

(Those comments have also been posted on the PLoS Computational Biology wiki.)

May I believe I am a Bayesian?!

“…the argument is false that because some ideal form of this approach to reasoning seems excellent n theory it therefore follows that in practice using this and only this approach to reasoning is the right thing to do.” Stephen Senn, 2011

Deborah Mayo, Aris Spanos, and Kent Staley have edited a special issue of Rationality, Markets and Morals (RMM) (a rather weird combination, esp. for a journal name!) on “Statistical Science and Philosophy of Science: Where Do (Should) They Meet in 2011 and Beyond?” for which comments are open. Stephen Senn has a paper therein entitled You May Believe You Are a Bayesian But You Are Probably Wrong in his usual witty, entertaining, and… Bayesian-bashing style! I find it very kind of him to allow us to remain in the wrong, very kind indeed…

   

Now, the paper somehow intersects with the comments Stephen made on our review of Harold Jeffreys’ Theory of Probability a while ago. It contains a nice introduction to the four great systems of statistical inference, embodied by de Finetti, Fisher, Jeffreys, and Neyman plus Pearson. The main criticism of Bayesianism à la de Finetti is that it is so perfect as to be outworldish. And, since this perfection is lost in the practical implementation, there is no compelling reason to be a Bayesian. Worse, that all practical Bayesian implementations conflict with Bayesian principles. Hence a Bayesian author “in practice is wrong”. Stephen concludes with a call for eclecticism, quite in line with his usual style since this is likely to antagonise everyone. (I wonder whether or not having no final dot to the paper has a philosophical meaning. Since I have been caught in over-interpreting book covers, I will not say more!) As I will try to explain below, I believe Stephen has paradoxically himself fallen victim of over-theorising/philosophising! (Referring the interested reader to the above post as well as to my comments on Don Fraser’s “Is Bayes posterior quick and dirty confidence?” for more related points. Esp. about Senn’s criticisms of objective Bayes on page 52 that are not so central to this discussion… Same thing for the different notions of probability [p.49] and the relative difficulties of the terms in (2) [p.50]. Deborah Mayo has a ‘deconstructed” version of Stephen’s paper on her blog, with a much deeper if deBayesian philosophical discussion. And then Andrew Jaffe wrote a post in reply to Stephen’s paper. Whose points I cannot discuss for lack of time, but with an interesting mention of Jaynes as missing in Senn’s pantheon.)

  

“The Bayesian theory is a theory on how to remain perfect but it does not explain how to become good.” Stephen Senn, 2011

While associating theories with characters is a reasonable rethoretical device, especially with large scale characters as the one above!, I think it deters the reader from a philosophical questioning on the theory behind the (big) man. (In fact, it is a form of bullying or, more politely (?), of having big names shoved down your throat as a form of argument.)  In particular, Stephen freezes the (Bayesian reasoning about the) Bayesian paradigm in its de Finetti phase-state, arguing about what de Finetti thought and believed. While this is historically interesting, I do not see why we should care at the praxis level. (I have made similar comments on this blog about the unpleasant aspects of being associated with one character, esp. the mysterious Reverent Bayes!) But this is not my main point.

“…in practice things are not so simple.” Stephen Senn, 2011

The core argument in Senn’s diatribe is that reality is always more complex than the theory allows for and thus that a Bayesian has to compromise on her/his perfect theory with reality/practice in order to reach decisions. A kind of philosophical equivalent to Achille and the tortoise. However, it seems to me that the very fact that the Bayesian paradigm is a learning principle implies that imprecisions and imperfections are naturally endowed into the decision process. Thus avoiding the apparent infinite regress (Regress ins Unendliche) of having to run a Bayesian analysis to derive the prior for the Bayesian analysis at the level below (which is how I interpret Stephen’s first paragraph in Section 3). By refusing the transformation of a perfect albeit ideal Bayesian into a practical if imperfect bayesian (or coherent learner or whatever name that does not sound like being a member of a sect!), Stephen falls short of incorporating the contrainte de réalité into his own paradigm. The further criticisms found about prior justification, construction, evaluation (pp.59-60) are also of that kind, namely preventing the statistician to incorporate a degree of (probabilistic) uncertainty into her/his analysis.

In conclusion, reading Stephen’s piece was a pleasant and thought-provoking moment. I am glad to be allowed to believe I am a Bayesian, even though I do not believe it is a belief! The praxis of thousands of scientists using Bayesian tools with their personal degree of subjective involvement is an evolutive organism that reaches much further than the highly stylised construct of de Finetti (or of de Finetti restaged by Stephen!). And appropriately getting away from claims to being perfect or right. Or even being more philosophical.

Bayesian ideas and data analysis

Here is [yet!] another Bayesian textbook that appeared recently. I read it in the past few days and, despite my obvious biases and prejudices, I liked it very much! It has a lot in common (at least in spirit) with our Bayesian Core, which may explain why I feel so benevolent towards Bayesian ideas and data analysis. Just like ours, the book by Ron Christensen, Wes Johnson, Adam Branscum, and Timothy Hanson is indeed focused on explaining the Bayesian ideas through (real) examples and it covers a lot of regression models, all the way to non-parametrics. It contains a good proportion of WinBugs and R codes. It intermingles methodology and computational chapters in the first part, before moving to the serious business of analysing more and more complex regression models. Exercises appear throughout the text rather than at the end of the chapters. As the volume of their book is more important (over 500 pages), the authors spend more time on analysing various datasets for each chapter and, more importantly, provide a rather unique entry on prior assessment and construction. Especially in the regression chapters. The author index is rather original in that it links the authors with more than one entry to the topics they are connected with (Ron Christensen winning the game with the highest number of entries).