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.