After a very, very long delay, we eventually re-revised our paper about necessary and sufficient conditions on summary statistics to be relevant for model choice (i.e. to lead to consistent tests). Reasons, both good and bad, abound for this delay! Some (rather bad) were driven by the completion of a certain new edition… Some (fairly good) are connected with the requests from the Series B editorial team, towards improving our methodological input. As a result we put more emphasis on the post-ABC cross-checking for the relevance of the summary choice, via a predictive posterior evaluation of the means of the summary statistic under both models and a test for mean equality. And re-ran a series of experiments on a three population population genetic example. Plus, on the side, simplified some of our assumptions. I dearly hope the paper can make it through but am also looking forward the opinion of the Series B editorial team The next version of Relevant statistics for Bayesian model choice should be arXived by now (meaning when this post appears!).
Archive for Pierre Simon de Laplace
This morning I gave my talk on ABC; computation or inference? at the appliBUGS seminar. Here, in Paris, BUGS stands for Bayesian United Group of Statisticians! Presumably in connection with a strong football culture, since the talk after mine was Jean-Louis Foulley’s ranking of the Euro 2012 teams. Quite an interesting talk (even though I am not particularly interested in football and even though I dozed a little, steaming out the downpour I had received on my bike-ride there…) I am also sorry I missed the next talk by Jean-Louis on Galton’s quincunx. (
Unfortunately, his slides are not [yet?] on-line.)
As a coincidence, after launching a BayesComp page on Google+ (as an aside, I am quite nonplussed by the purpose of Google-), Nicolas Chopin also just started a Bayes in Paris webpage, in connection with our informal seminar/reading group at CREST. With the appropriate picture this time, i.e. a street plaque remembering…Laplace! May I suggest the RER stop Laplace and his statue in the Paris observatory as additional illustrations for the other pages…
No, no, this is not an announcement for a meeting on an Australian beach (which is Bayes on the Beach, taking place next November (6-8) on the Sunshine Coast and is organised by Kerrie Mengersen’s BRAG, at QUT, that I just left! With Robert Wolpert as the international keynote speaker and Matt Wand as the Australian keynote speaker.) Bayes by the Bay is “a pedagogical workshop on Bayesian methods in Science” organised by the Institute of Mathematical Sciences, based in the CIT campus in Chennai. It is taking place on January 4-8, 2013, in Pondichéry. (To use the French spelling of this former comptoir of French India…) Just prior to the ISBA Varanasi meeting on Bayesian Statistics.
Great: the webpage for the workshop uses the attached picture of Pierre-Simon (de) Laplace, rather than the unlikely picture of Thomas Bayes found all over the place (incl. this blog!). This was also the case in Christensen et al.’s Bayesian ideas and data analysis. So maybe there is a trend there. I also like the name “Bayes by the Bay“, as it reminds me of a kid song we used to sing to/with our kids when they were young, “down by the bay“, after a summer vacation with Anne and George Casella…
The debate about non-informative priors has been going on for ages, at least since the end of the 19th century with criticisms by Bertrand and de Morgan about the lack of invariance of Laplace’s uniform priors (the same criticism reported by Stéphane Laurent in the above comments). This lack of invariance sounded like a death stroke for the Bayesian approach and, while some Bayesians were desperately trying to cling to specific distributions, using less-than-formal arguments, others had a wider vision of a larger picture where priors could be used in situations where there was hardly any prior information, beyond the shape of the likelihood itself. (This was even before Abraham Wald established his admissibility and complete class results about Bayes procedures. And at about the same time as E.J.G. Pitman gave an “objective” derivation of the best invariant estimator as a Bayes estimator against the corresponding Haar measure…)
This vision is best represented by Jeffreys’ distributions, where the information matrix of the sampling model, , is turned into a prior distribution
which is most often improper, i.e. does not integrate to a finite value. The label “non-informative” associated with Jeffreys’ priors is rather unfortunate, as they represent an input from the statistician, hence are informative about something! Similarly, “objective” has an authoritative weight I dislike… I thus prefer the label “reference prior”, used for instance by José Bernado.
Those priors indeed give a reference against which one can compute either the reference estimator/test/prediction or one’s own estimator/test/prediction using a different prior motivated by subjective and objective items of information. To answer directly the question, “why not use only informative priors?”, there is actually no answer. A prior distribution is a choice made by the statistician, neither a state of Nature nor a hidden variable. In other words, there is no “best prior” that one “should use”. Because this is the nature of statistical inference that there is no “best answer”.
Hence my defence of the noninformative/reference choice! It is providing the same range of inferential tools as other priors, but gives answers that are only inspired by the shape of the likelihood function, rather than induced by some opinion about the range of the unknown parameters.
The Brazilian society for Bayesian Analysis (ISBrA, whose annual meeting is taking place at this very time!) asked me to write a review on Pierre Simon Laplace’s book, Théorie Analytique des Probabilités, a book that was initially published in 1812, exactly two centuries ago. I promptly accepted this request as (a) I had never looked at this book and so this provided me with a perfect opportunity to do so, (b) while in Vancouver, Julien Cornebise had bought for me a 1967 reproduction of the 1812 edition, (c) I was curious to see how much of the book had permeated modern probability and statistics or, conversely, how much of Laplace’s perspective was still understandable by modern day standards. (Note that the link on the book leads to a free version of the 1814, not 1812, edition of the book, as free as the kindle version on amazon.)
“Je m’attache surtout, à déterminer la probabilité des causes et des résultats indiqués par événemens considérés en grand nombre.” P.S. Laplace, Théorie Analytique des Probabilités, page 3
First, I must acknowledge I found the book rather difficult to read and this for several reasons: (a) as is the case for books from older times, the ratio text-to-formulae is very high, with an inconvenient typography and page layout (ar least for actual standards), so speed-reading is impossible; (b) the themes offered in succession are often abruptly brought and uncorrelated with the previous ones; (c) the mathematical notations are 18th-century, so sums are indicated by S, exponentials by c, and so on, which again slows down reading and understanding; (d) for all of the above reasons, I often missed the big picture and got mired into technical details until they made sense or I gave up; (e) I never quite understood whether or not Laplace was interested in the analytics like generating functions only to provide precise numerical approximations or for their own sake. Hence a form of disappointment by the end of the book, most likely due to my insufficient investment in the project (on which I mostly spent an Amsterdam/Calgary flight and jet-lagged nights at BIRS…), even though I got excited by finding the bits and pieces about Bayesian estimation and testing. Continue reading