whetstone and alum block for Occam’s razor

A strange title, if any! (The whetstone is a natural hard stone used for sharpening steel instruments, like knifes or sickles and scythes, I remember my grand-fathers handling one when cutting hay and weeds. Alum is hydrated potassium aluminium sulphate and is used as a blood coagulant. Both items are naturally related with shaving and razors, if not with Occam!) The whole title of the paper published by Guido Consonni, Jon Forster and Luca La Rocca in Statistical Science is “The whetstone and the alum block: balanced objective Bayesian comparison of nested models for discrete data“. The paper builds on the notions introduced in the last Valencia meeting by Guido and Luca (and discussed by Judith Rousseau and myself).

Beyond the pun (that forced me to look for “alum stone” on Wikipedia!, and may be lost on some other non-native readers), the point in the title is to build a prior distribution aimed at the comparison of two models such that those models are more sharply distinguished: Occam’s razor would thus cut better when the smaller model is true (hence the whetstone) and less when it is not (hence the alum block)… The solution proposed by the authors is to replace the reference prior on the larger model, π₁, with a moment prior à la Johnson and Rossell (2010, JRSS B) and then to turn this moment prior into an intrinsic prior à la Pérez and Berger (2002, Biometrika), making it an “intrinsic moment”. The first transform turns π₁ into a non-local prior, with the aim of correcting for the imbalanced convergence rates of the Bayes factor under the null and under the alternative (this is the whetstone). The second transform accumulates more mass in the vicinity of the null model (this is the alum block). (While I like the overall perspective on intrinsic priors, the introduction is a wee confusing about them, e.g. when it mentions fictive observations instead of predictives.)

Being a referee for this paper, I read it in detail (and also because this is one of my favourite research topics!) Further, we already engaged into a fruitful discussion with Guido since the last Valencia meeting and the current paper incorporates some of our comments (and replies to others). I find the proposal of the authors clever and interesting, but not completely Bayesian. Overall, the paper provides a clearly novel methodology that calls for further studies…