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.

Bayesian computational tools

I just arXived a survey entitled Bayesian computational tools in connection with a chapter the editors of the Annual Review of Statistics and Its Application asked me to write. (A puzzling title, I would have used Applications, not Application. Puzzling journal too: endowed with a prestigious editorial board, I wonder at the long-term perspectives of the review, once “all” topics have been addressed. At least, the “non-profit” aspect is respected: $100 for personal subscriptions and $250 for libraries, plus a one-year complimentary online access to volume 1.) Nothing terribly novel in my review, which illustrates some computational tool in some Bayesian settings, missing five or six pages to cover particle filters and sequential Monte Carlo. I however had fun with a double-exponential (or Laplace) example. This distribution indeed allows for a closed-form posterior distribution on the location parameter under a normal prior, which can be expressed as a mixture of truncated normal distributions. A mixture of (n+1) normal distributions for a sample of size n. We actually noticed this fact (which may already be well-known) when looking at our leading example in the consistent ABC choice paper, but it vanished from the appendix in the later versions. As detailed in the previous post, I also fought programming issues induced by this mixture, due to round-up errors in the most extreme components, until all approaches provided similar answers.

re-re-relevant statistics for ABC model choice

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!).

Glasgow bridges & talk

I had a nice run along the Kelvin river in Glasgow this morning, passing an incredible number of old bridges, some of them derelict with trees growing on them… Then came back to reorganise my slides towards a better introduction to ABC and a faster focus on the consistency result and assessment.

potentially relevant

This week, freshly back from Roma, I got the reviews on our paper “Relevant statistics for Bayesian model choice” from Series B. The comments are detailed and mostly to the point, expressing concern about the relevance of the paper for statistical methodology as the major issue.  We are thus asked for a revision making a much better connection with ABC methodology.

This is not an unexpected outcome, from my point of view, because the paper is indeed quite theoretical and the mathematical assumptions required to obtain the convergence theorems are rather overwhelming… Meaning that in practical cases they cannot truly be checked. However, I think we can eventually address those concerns for two distinct reasons: first, the paper comes as a third step in a series of papers where we first identified a sufficiency property, then realised that this property was actually quite a rare occurrence, and finally made a theoretical advance as to when is a summary statistic enough (i.e. “sufficient” in the standard sense of the term!)  to conduct model choice, with a clear answer that the mean ranges of the summary statistic under each model could not intersect.  Second, my own personal view is that those assumptions needed for convergence are not of the highest importance for statistical practice (even though they are needed in the paper!) and thus that, from a methodological point of view, only the conclusion should be taken into account. It is then rather straightforward to come up with (quick-and-dirty) simulation devices to check whether a summary statistic behaves differently under both models, taking advantage of the reference table already available (instead of having to run Monte Carlo experiments with ABC basis)…

One of the comments was that maybe Bayes factors were not appropriate for conducting model choice, thus making the whole derivation irrelevant. This is a possible perspective but it can be objected that Bayes factors and posterior probabilities are used in conjunction with ABC in dozens of genetic papers. Further arguments are provided in the various replies to both of Templeton’s radical criticisms. That more empirical and model-based assessments also are available is quite correct, as demonstrated in the multicriterion approach of Olli Ratmann and co-authors. This is simply another approach, not followed by most geneticists so far…