Archive for Bayesian model choice

comparison of Bayesian predictive methods for model selection

Posted in Books, Statistics, University life with tags , , , , , , , , , on April 9, 2015 by xi'an

“Dupuis and Robert (2003) proposed choosing the simplest model with enough explanatory power, for example 90%, but did not discuss the effect of this threshold for the predictive performance of the selected models. We note that, in general, the relative explanatory power is an unreliable indicator of the predictive performance of the submodel,”

Juho Piironen and Aki Vehtari arXived a survey on Bayesian model selection methods that is a sequel to the extensive survey of Vehtari and Ojanen (2012). Because most of the methods described in this survey stem from Kullback-Leibler proximity calculations, it includes some description of our posterior projection method with Costas Goutis and Jérôme Dupuis. We indeed did not consider prediction in our papers and even failed to include consistency result, as I was pointed out by my discussant in a model choice meeting in Cagliari, in … 1999! Still, I remain fond of the notion of defining a prior on the embedding model and of deducing priors on the parameters of the submodels by Kullback-Leibler projections. It obviously relies on the notion that the embedding model is “true” and that the submodels are only approximations. In the simulation experiments included in this survey, the projection method “performs best in terms of the predictive ability” (p.15) and “is much less vulnerable to the selection induced bias” (p.16).

Reading the other parts of the survey, I also came to the perspective that model averaging makes much more sense than model choice in predictive terms. Sounds obvious stated that way but it took me a while to come to this conclusion. Now, with our mixture representation, model averaging also comes as a natural consequence of the modelling, a point presumably not stressed enough in the current version of the paper. On the other hand, the MAP model now strikes me as artificial and linked to a very rudimentary loss function. A loss that does not account for the final purpose(s) of the model. And does not connect to the “all models are wrong” theorem.

nested sampling for systems biology

Posted in Books, Statistics, University life with tags , , , , on January 14, 2015 by xi'an

In conjunction with the recent PNAS paper on massive model choice, Rob Johnson†, Paul Kirk and Michael Stumpf published in Bioinformatics an implementation of nested sampling that is designed for biological applications, called SYSBIONS. Hence the NS for nested sampling! The C software is available on-line. (I had planned to post this news next to my earlier comments but it went under the radar…)

full Bayesian significance test

Posted in Books, Statistics with tags , , , , , , , , , , on December 18, 2014 by xi'an

Among the many comments (thanks!) I received when posting our Testing via mixture estimation paper came the suggestion to relate this approach to the notion of full Bayesian significance test (FBST) developed by (Julio, not Hal) Stern and Pereira, from São Paulo, Brazil. I thus had a look at this alternative and read the Bayesian Analysis paper they published in 2008, as well as a paper recently published in Logic Journal of IGPL. (I could not find what the IGPL stands for.) The central notion in these papers is the e-value, which provides the posterior probability that the posterior density is larger than the largest posterior density over the null set. This definition bothers me, first because the null set has a measure equal to zero under an absolutely continuous prior (BA, p.82). Hence the posterior density is defined in an arbitrary manner over the null set and the maximum is itself arbitrary. (An issue that invalidates my 1993 version of the Lindley-Jeffreys paradox!) And second because it considers the posterior probability of an event that does not exist a priori, being conditional on the data. This sounds in fact quite similar to Statistical Inference, Murray Aitkin’s (2009) book using a posterior distribution of the likelihood function. With the same drawback of using the data twice. And the other issues discussed in our commentary of the book. (As a side-much-on-the-side remark, the authors incidentally  forgot me when citing our 1992 Annals of Statistics paper about decision theory on accuracy estimators..!)

postdoc in Paris?

Posted in Kids, Statistics, Travel, University life with tags , , , , , , , on November 4, 2014 by xi'an

Pont Alexandre III, Paris, May 8, 2012. On our way to the old-fashioned science museum, Palais de la Découverte, we had to cross the bridge on foot as the nearest métro station was closed, due to N. Sarkozy taking part in a war memorial ceremony there...There is an open call of the Fondation Sciences Mathématiques de Paris (FSMP) about a postdoctoral funding program with 18 position-years available for staying in Université Paris-Dauphine (and other participating universities). The net support is quite decent  (wrt French terms and academic salaries) and the application form easy to fill. So, if you are interested in coming to Paris to work on ABC, MCMC, Bayesian model choice, &tc., feel free to contact me (or another Parisian statistician) and to apply! The deadline is December 01, 2014.  And the decision will be made by January 15, 2015. The starting date for the postdoc is October 01, 2015.

all models are wrong

Posted in Statistics, University life with tags , , , , , , , on September 27, 2014 by xi'an

“Using ABC to evaluate competing models has various hazards and comes with recommended precautions (Robert et al. 2011), and unsurprisingly, many if not most researchers have a healthy scepticism as these tools continue to mature.”

Michael Hickerson just published an open-access letter with the above title in Molecular Ecology. (As in several earlier papers, incl. the (in)famous ones by Templeton, Hickerson confuses running an ABC algorithm with conducting Bayesian model comparison, but this is not the main point of this post.)

“Rather than using ABC with weighted model averaging to obtain the three corresponding posterior model probabilities while allowing for the handful of model parameters (θ, τ, γ, Μ) to be estimated under each model conditioned on each model’s posterior probability, these three models are sliced up into 143 ‘submodels’ according to various parameter ranges.”

The letter is in fact a supporting argument for the earlier paper of Pelletier and Carstens (2014, Molecular Ecology) which conducted the above splitting experiment. I could not read this paper so cannot judge of the relevance of splitting this way the parameter range. From what I understand it amounts to using mutually exclusive priors by using different supports.

“Specifically, they demonstrate that as greater numbers of the 143 sub-models are evaluated, the inference from their ABC model choice procedure becomes increasingly.”

An interestingly cut sentence. Increasingly unreliable? mediocre? weak?

“…with greater numbers of models being compared, the most probable models are assigned diminishing levels of posterior probability. This is an expected result…”

True, if the number of models under consideration increases, under a uniform prior over model indices, the posterior probability of a given model mechanically decreases. But the pairwise Bayes factors should not be impacted by the number of models under comparison and the letter by Hickerson states that Pelletier and Carstens found the opposite:

“…pairwise Bayes factor[s] will always be more conservative except in cases when the posterior probabilities are equal for all models that are less probable than the most probable model.”

Which means that the “Bayes factor” in this study is computed as the ratio of a marginal likelihood and of a compound (or super-marginal) likelihood, averaged over all models and hence incorporating the prior probabilities of the model indices as well. I had never encountered such a proposal before. Contrary to the letter’s claim:

“…using the Bayes factor, incorporating all models is perhaps more consistent with the Bayesian approach of incorporating all uncertainty associated with the ABC model choice procedure.”

Besides the needless inclusion of ABC in this sentence, a somewhat confusing sentence, as Bayes factors are not, stricto sensu, Bayesian procedures since they remove the prior probabilities from the picture.

“Although the outcome of model comparison with ABC or other similar likelihood-based methods will always be dependent on the composition of the model set, and parameter estimates will only be as good as the models that are used, model-based inference provides a number of benefits.”

All models are wrong but the very fact that they are models allows for producing pseudo-data from those models and for checking if the pseudo-data is similar enough to the observed data. In components that matters the most for the experimenter. Hence a loss function of sorts…

JSM 2014, Boston [#3]

Posted in Statistics, University life with tags , , , , , , , on August 8, 2014 by xi'an

Today I gave a talk in the Advances in model selection session. Organised by Veronika Rockova and Ed George. (A bit of pre-talk stress: I actually attempted to change my slides at 5am and only managed to erase the current version! I thus left early enough to stop by the presentation room…) Here are the final slides, which have much in common with earlier versions, but also borrowed from Jean-Michel Marin’s talk in Cambridge. A posteriori, I think the talk missed one slide on the practical run of the ABC random forest algorithm, since later questions showed miscomprehension from the audience.

The other talks in this session were by Andreas Buja [whom I last met in Budapest last year] on valid post-modelling inference. A very relevant reflection on the fundamental bias in statistical modelling. Then by Nick Polson, about efficient ways to compute MAP for objective functions that are irregular.  Great entry into optimisation methods I had never heard of earlier.! (The abstract is unrelated.) And last but not least by Veronika Rockova, on mixing Indian buffet processes with spike-and-slab priors for factor analysis with unknown numbers of factors. A definitely advanced contribution to factor analysis, with a very nice idea of introducing a non-identifiable rotation to align on orthogonal designs. (Here too the abstract is unrelated, a side effect of the ASA requiring abstracts sent very long in advance.)

Although discussions lasted well into the following Bayesian Inference: Theory and Foundations session, I managed to listen to a few talks there. In particular, a talk by Keli Liu on constructing non-informative priors. A question of direct relevance. The notion of objectivity is to achieve a frequentist distribution of the Bayes factor associated with the point null that is constant. Or has a constant quantile at a given level. The second talk by Alexandra Bolotskikh related to older interests of mine’s, namely the construction of improved confidence regions in the spirit of Stein. (Not that surprising, given that a coauthor is Marty Wells, who worked with George and I on the topic.) A third talk by Abhishek Pal Majumder (jointly with Jan Hanning) dealt on a new type of fiducial distributions, with matching prior properties. This sentence popped a lot over the past days, but this is yet another area where I remain puzzled by the very notion. I mean the notion of fiducial distribution. Esp. in this case where the matching prior gets even closer to being plain Bayesian.

R/Rmetrics in Paris [alas!]

Posted in Mountains, pictures, R, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , on June 30, 2014 by xi'an

Bernard1Today I gave a talk on Bayesian model choice in a fabulous 13th Century former monastery in the Latin Quarter of Paris… It is the Collège des Bernardins, close to Jussieu and Collège de France, unbelievably hidden to the point I was not aware of its existence despite having studied and worked in Jussieu since 1982… I mixed my earlier San Antonio survey on importance sampling approximations to Bayes factors with an entry to our most recent work on ABC with random forests. This was the first talk of the 8th R/Rmetrics workshop taking place in Paris this year. (Rmetrics is aiming at aggregating R packages with econometrics and finance applications.) And I had a full hour and a half to deliver my lecture to the workshop audience. Nice place, nice people, new faces and topics (and even andouille de Vire for lunch!): why should I complain with an alas in the title?!Bernard2What happened is that the R/Rmetrics meetings have been till this year organised in Meielisalp, Switzerland. Which stands on top of Thuner See and… just next to the most famous peaks of the Bernese Alps! And that I had been invited last year but could not make it… Meaning I lost a genuine opportunity to climb one of my five dream routes, the Mittelegi ridge of the Eiger. As the future R/Rmetrics meetings will not take place there.

A lunch discussion at the workshop led me to experiment the compiler library in R, library that I was unaware of. The impact on the running time is obvious: recycling the fowler function from the last Le Monde puzzle,

> bowler=cmpfun(fowler)
> N=20;n=10;system.time(fowler(pred=N))
   user  system elapsed 
 52.647   0.076  56.332 
> N=20;n=10;system.time(bowler(pred=N))
   user  system elapsed 
 51.631   0.004  51.768 
> N=20;n=15;system.time(bowler(pred=N))
   user  system elapsed 
 51.924   0.024  52.429 
> N=20;n=15;system.time(fowler(pred=N))
   user  system elapsed 
 52.919   0.200  61.960 

shows a ten- to twenty-fold gain in system time, if not in elapsed time (re-alas!).

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