proper likelihoods for Bayesian analysis

While in Montpellier yesterday (where I also had the opportunity of tasting an excellent local wine!), I had a look at the 1992 Biometrika paper by Monahan and Boos on “Proper likelihoods for Bayesian analysis“. This is a paper I missed and that was pointed out to me during the discussions in Padova. The main point of this short paper is to decide when a method based on an approximative likelihood function is truly (or properly) Bayes. Just the very question a bystander would ask of ABC methods, wouldn’t it?! The validation proposed by Monahan and Boos is one of calibration of credible sets, just as in the recent arXiv paper of Dennis Prangle, Michael Blum, G. Popovic and Scott Sisson I reviewed three months ago. The idea is indeed to check by simulation that the true posterior coverage of an α-level set equals the nominal coverage α. In other words, the predictive based on the likelihood approximation should be uniformly distributed and this leads to a goodness-of-fit test based on simulations. As in our ABC model choice paper, Proper likelihoods for Bayesian analysis notices that Bayesian inference drawn upon an insufficient statistic is proper and valid, simply less accurate than the Bayesian inference drawn upon the whole dataset. The paper also enounces a conjecture:

A [approximate] likelihood L is a coverage proper Bayesian likelihood if and inly if L has the form L(y|θ) = c(s) g(s|θ) where s=S(y) is a statistic with density g(s|θ) and c(s) some function depending on s alone.

conjecture that sounds incorrect in that noisy ABC is also well-calibrated. (I am not 100% sure of this argument, though.) An interesting section covers the case of pivotal densities as substitute likelihoods and of the confusion created by the double meaning of the parameter θ. The last section is also connected with ABC in that Monahan and Boos reflect on the use of large sample approximations, like normal distributions for estimates of θ which are a special kind of statistics, but do not report formal results on the asymptotic validation of such approximations. All in all, a fairly interesting paper!

Reading this highly interesting paper also made me realise that the criticism I had made in my review of Prangle et al. about the difficulty for this calibration method to address the issue of summary statistics was incorrect: when using the true likelihood function, the use of an arbitrary summary statistics is validated by this method and is thus proper.

Core [still] minus one…

Another full day spent working with Jean-Michel Marin on the new edition of Bayesian Core (soon to be Bayesian Essentials with R!) and the remaining hierarchical Bayes chapter… I have reread and completed the regression and GLM chapters, sent to very friendly colleagues for a last round of comments. Now, I am essentially idle, waiting for Jean-Michel to finish his part on the hierarchical Bayes chapter, so that I can do the final editing.round. Jean-Michel had a very long day on that chapter, leaving Montpellier at 5am to return only at half past midnight, due to massive delays in the train schedule (which is why I always fly to Montpellier…)

Core minus one!

Jean-Michel Marin visited me in Paris last week and, besides taking part in Pierre’s PhD defence, we made enough progress to close two more chapters of the new edition of Bayesian Core (soon to be Bayesian Essentials with R!) This follows the good work session we had in Carnon where we also completed two chapters (although it was hard to convince anyone that renting a flat by the Mediterranean sea was at all connected with…work! While it was: the only breaks I took were my morning runs…). There just remains one single chapter to complete, now, the one on hierarchical Bayes models. By all means, I dearly want to see it done by November 1!!!

NYT persp’ on Melbourne

I came upon this New York Times argument for placing Melbourne in #15 among the 41 places to go in 2011:

With a bunch of new hotels and restaurants led by notable chefs cropping up, Melbourne has been stealing the spotlight from its sister city, Sydney. The most notable addition comes from the luxury brand Crown, which is investing 1 billion Australian dollars (about the same in U.S. dollars) to expand its sprawling Crown Entertainment Complex on the southern bank of the Yarra River. In April it opened Australia’s largest hotel, the 300-million-dollar 658-room Crown Metropol, which has an infinity pool on the 27th floor with 180-degree views of the city, and is home to the Maze and Maze Grill, the celebrity chef Gordon Ramsay’s first endeavors Down Under. The complex also includes the Crown Towers hotel, which has four private penthouse gaming salons with 360-degree views of Melbourne’s skyline.

The city’s thriving arts scene now has stylish boutique hotels to match, too. Three Art Series Hotels, inspired by (and featuring the works of) famous artists, opened in the last year. The Olsen, named for the landscape painter John Olsen, is the flagship of the group, with 229 rooms (from 215 dollars a night) and a heated, glass-bottomed swimming pool.

Visiting foodies will be able to choose from a number of new restaurants. In October, the Australian chef Neil Perry, of Rockpool in Sydney, opened Spice Temple, a 200-seat contemporary Szechuan restaurant next door to his Rockpool Bar & Grill in the Crown complex, as well as a new bar, the Waiting Room, in the lobby of the Crown Towers hotel. Also within the Crown complex, a new seafood restaurant, the Atlantic, will debut in February with Donovan Cooke as executive chef.

This is fairly puzzling, Not the fact that Melbourne is on the list, of course, this is indeed an attractive and thriving city I enjoyed living in the past two weeks. But the reasons provided here are just so unappealing. A new expensive hotel? Duh.  A new restaurant? Doh. (Plus, there already is a highly rated Spice Temple in Sydney! Why bother with a replica?) Reading through the series with a new eye makes me seriously wonder if this is anything else but covert advertising… (In the 2012 version of this NYT list, Montpellier appears as the French entry…not for its beautiful medieval centre but for its modern architecture and for its tramway, which has been completed but which construction created such a traffic nightmare over the years I have visited Jean-Michel Marin there.)

Étang du Grec, Carnon