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.
Archive for ABC model choice
Here are the slides of my talk in Padova for the workshop Recent Advances in statistical inference: theory and case studies (very similar to the slides for the Varanasi and Gainesville meetings, obviously!, with Peter Müller commenting [at last!] that I had picked the wrong photos from Khajuraho!)
The worthy Padova addendum is that I had two discussants, Stefano Cabras from Universidad Carlos III in Madrid, whose slides are :
and Francesco Pauli, from Trieste, whose slides are:
These were kind and rich discussions with many interesting openings: Stefano’s idea of estimating the pivotal function h is opening new directions, obviously, as it indicates an additional degree of freedom in calibrating the method. Esp. when considering the high variability of the empirical likelihood fit depending on the the function h. For instance, one could start with a large collection of candidate functions and build a regression or a principal component reparameterisation from this collection… (Actually I did not get point #1 about ignoring f: the empirical likelihood is by essence ignoring anything outside the identifying equation, so as long as the equation is valid..) Point #2: Opposing sample free and simulation free techniques is another interesting venue, although I would not say ABC is “sample free”. As to point #3, I will certainly get a look at Monahan and Boos (1992) to see if this can drive the choice of a specific type of pseudo-likelihoods. I like the idea of checking the “coverage of posterior sets” and even more “the likelihood must be the density of a statistic, not necessarily sufficient” as it obviously relates with our current ABC model comparison work… Esp. when the very same paper is mentioned by Francesco as well. Grazie, Stefano! I also appreciate the survey made by Francesco on the consistency conditions, because I think this is an important issue that should be taken into consideration when designing ABC algorithms. (Just pointing out again that, in the theorem of Fearnhead and Prangle (2012) quoting Bernardo and Smith (1992), some conditions are missing for the mathematical consistency to apply.) I also like the agreement we seem to reach about ABC being evaluated per se rather than an a poor man’s Bayesian method. Francesco’s analysis of Monahan and Boos (1992) as validating or not empirical likelihood points out a possible link with the recent coverage analysis of Prangle et al., discussed on the ‘Og a few weeks ago. And an unsuspected link with Larry Wasserman! Grazie, Francesco!
Our paper on using empirical likelihood for Bayesian computation (with Kerrie Mengersen and Pierre Pudlo) has been accepted by PNAS [after we removed the A from ABCel!], which is terrific news! It has already appeared on-line as early edition in the issue of January 7. Which is also terrific! (Unfortunately, it is not open access, contrary to the previous PNAS paper on ABC model choice as the cost was just too high.)
As mentioned in my review of Paradoxes in Scientific Inference I was a bit confused by this presentation of the likelihood principle and this led me to ponder for a week or so whether or not there was an issue with Birnbaum’s proof (or, much more likely, with my vision of it!). After reading again Birnbaum’s proof, while sitting down in a quiet room at ICERM for a little while, I do not see any reason to doubt it. (Keep reading at your own risk!)
My confusion was caused by mixing sufficiency in the sense of Birnbaum’s mixed experiment with sufficiency in the sense of our ABC model choice PNAS paper, namely that sufficient statistics are not always sufficient to select the right model. The sufficient statistics in the proof reduces the (2,x2) observation from Model 2 to (1,x1) from Model 1 when there is an observation x1 that produces a likelihood proportional to the likelihood for x2 and the statistic is indeed sufficient: the distribution of (2,x2) given (1,x1) does not depend on the parameter θ. Of course, the statistic is not sufficient (most of the time) for deciding between Model 1 and Model 2, but this model choice issue is foreign to Birnbaum’s construction.
Next week, I will visit both Iowa State University, in Ames—a funny item for French speaking readers is that I will first land in Des Moines before reaching (les) Ames!, a logical step if any, even though only the first name relates to the early French exploration of the area: Ames has apparently no [ethymological] connection with souls…—, and the University of Chicago Booth Business School, giving a seminar on ABC model choice and empirical likelihood in both places. (I have never been to Iowa before and the last time I visited Chicago—rather than just commuting through O’Hare—was in May 1988, when I drove a friend to the airport…!) Here are the time and places for the seminars (note that the seminar at Booth is on Tuesday rather than on the customary Thursday to accommodate my tight schedule!):
- Monday Oct 29th, 4:10 pm, Snedecor 3105, Iowa State University, Zyskind lecture [abstract]
- Tuesday Oct 30th, 12:00pm, Lunch: C50B and 1:20pm Seminar: C50A, the University of Chicago Booth Business School, Econometrics and Statistics Colloquium [abstract]
As a coincidence—not so much as he is currently assistant professor in Ames—, the previous seminar speaker in Ames is my friend Vivek Roy, talking on Monte Carlo Methods for Improper Target Distributions! Here is (again!) the current version of the slides: