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.