Here are the slides for my talks both at CREST this afternoon (in ½ an hour!) and in Madrid [on Friday 11/11/11=16, magical day of the year, especially since I will be speaking at 11:11 CET...] for the Workshop Métodos Bayesianos 11 (no major difference with the slides from Zürich, hey!, except for the quantile distribution example]
Archive for sufficient statistics
Here are the slides of my talk yesterday at the Computational Methods in Applied Sciences workshop in Columbia:
The last section of the talk covers our new results with Jean-Michel Marin, Natesh Pillai and Judith Rousseau on the necessary and sufficient conditions for a summary statistic to be used in ABC model choice. (The paper is about to be completed.) This obviously comes as the continuation of our reflexions on ABC model choice started last January. The major message of the paper is that the statistics used for running model choice cannot have a mean value common to both models, which strongly implies using ancillary statistics with different means under each model. (I am afraid that, thanks to the mixture of no-jetlag fatigue and of slide inflation [95 vs. 40mn] and of asymptotics technicalities in the last part, the talk was far from comprehensible. I started on the wrong foot with not getting an XL [Xiao-Li's] comment on the measure-theory problem with the limit in ε going to zero. A peak given that great debate we had in Banff with Jean-Michel, David Balding, and Mark Beaumont, years ago. And our more recent paper about the arbitrariness of the density value in the Savage-Dickey paradox. I then compounded the confusion by stating the empirical mean was sufficient in the Laplace case…which is not even an exponential family. I hope I will be more articulate next week in Zürich where at least I will not speak past my bedtime!)
Chris Barnes, Sarah Filippi, Michael P.H. Stumpf, and Thomas Thorne posted a paper on arXiv on the selection of sufficient statistics towards ABC model choice. This paper, called Considerate Approaches to Achieving Sufficiency for ABC model selection, was presented by Chris Barnes during ABC in London two months ago. (Note that all talks of the meeting are now available in Nature Precedings. A neat concept by the way!) This paper of them builds on our earlier warning about (unfounded) ABC model selection to propose a selection of summary statistics that partly alleviates the original problem. (The part about the discrepancy with the true posterior probability remains to be addressed. As does the issue of whether or not the selected collection of statistics provides a convergent model choice inference. We are currently working on it…) Their section “Resuscitating ABC model choice” states quite clearly the goal of the paper:
- this [use of inadequate summary statistics] mirrors problems that can also be observed in the parameter estimation context,
- for many important, and arguably the most important applications of ABC, this problem can in principle be avoided by using the whole data rather than summary statistics,
- in cases where summary statistics are required, we argue that we can construct approximately sufficient statistics in a disciplined manner,
- when all else fails, a change in perspective, allows us to nevertheless make use of the flexibility of the ABC framework
The driving idea in the paper is to use an entropy approximation to measure the lack of information due to the use of a given set of summary statistics. The corresponding algorithm then proceeds from a starting pool of summary statistics to build sequentially a collection of the most informative summary statistics (which, in a sense, reminded me of a variable selection procedure based on Kullback-Leibler, we developed with Costas Goutis and Jérôme Dupuis). It is a very interesting advance in the issue of ABC model selection, even though it cannot eliminate all stumbling blocks. The interpretation that ABC should be processed as an inferential method on its own rather than an approximation to Bayesian inference is clearly appealing. (Fearnhead and Prangle, and Dean, Singh, Jasra and Peters could be quoted as well.)
Following the comments on our earlier submission to PNAS, we have written (and re-arXived) a revised version where we try to spell out (better) the distinction between ABC point (and confidence) estimation and ABC model choice, namely that the problem was at another level for Bayesian model choice (using posterior probabilities). When doing point estimation with in-sufficient summary statistics, the information content is poorer, but unless one uses very degraded summary statistics, inference is converging. We completely agree with the reviewers that the posterior distribution is different from the true posterior in this case but, at least, gathering more observations brings more information about the parameter (and convergence when the number of observations goes to infinity). For model choice, this is not guaranteed if we use summary statistics that are not inter-model sufficient, as shown by the Poisson and normal examples. Furthermore, except for very specific cases such as Gibbs random fields, it is almost always impossible to derive inter-model sufficient statistics, beyond the raw sample. This is why we consider there is a fundamental difference between point estimation and model choice.
Following the request from a referee, we also ran a more extensive simulation experiment for comparing two scenarios with 3 populations, 100 diploid individuals per population, and 50 loci/markers. However, the results are somehow less conclusive, in the sense that, since we use 50 loci, the data is much more informative about the model and therefore both the importance sampling and the ABC approximations provide a value of the posterior probability approximation that is close to one, hence both concluding with the validation of the true model. Because both approximations are very close to one, it is difficult to assess the worth of the ABC approximation per se, i.e. in numerical terms. (The fact that the statistical conclusion is the same for both approaches is of course satisfying from an inferential perspective, but is an altogether separate issue from our argument about the possible lack of convergence of the ABC Bayes factor approximation to the true Bayes factor.) Furthermore, this experiment may be beyond the manageable/reasonable in the sense that the importance sampling approximation cannot be taken for granted, nor can it be checked empirically. Indeed, with 50 markers and 100 individuals, the product likelihood suffers from an enormous variability that 100,000 particles and 100 trees per locus have trouble to address (despite a huge computing cost of more than 12 days on a powerful cluster).
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but it vanisheds with the options
which is an easy fix.
On Friday, I received a nice but embarrassing email from Xavier Didelot. He indeed reminded me that I attended the talk he gave at the model choice workshop in Warwick last May, as, unfortunately but rather unsurprisingly giving my short span memory!, I had forgotten about it! Looking at the slides he joined to his email, I indeed remember attending the talk and expecting to get back to the results after the meeting. As I went from Warwick to Paris only to leave a day after for Benidorm, and the Valencia 9 meeting, in such a hurry that I even forgot my current black notebook, the plans of getting back to the talk got forgotten so completely that even reading the tech report (now appeared in Bayesian Analysis) could not rescind them!
Here are some of Xavier’s comments, followed by my answers: Read more »