**A**lthough it went completely out of my mind, thanks to a rather heavy travel schedule, I gave last week a short interview about the notion of mathematical models, which got broadcast this week on France Culture, one of the French public radio channels. Within the daily *La Méthode Scientifique* show, which is a one-hour emission on scientific issues, always a [rare] pleasure to listen to. (Including the day they invited Claire Voisin.) The theme of the show that day was about the unreasonable effectiveness of mathematics, with the [classical] questioning of whether it is an efficient tool towards solving scientific (and inference?) problems because the mathematical objects pre-existed their use *or* we are (pre-)conditioned to use mathematics to solve problems. I somewhat sounded like a dog in a game of skittles, but it was interesting to listen to the philosopher discussing my relativistic perspective [provided you understand French!]. And I appreciated very much the way Céline Loozen the journalist who interviewed me sorted the chaff from the wheat in the original interview to make me sound mostly coherent! (A coincidence: Jean-Michel Marin got interviewed this morning on France Inter, the major public radio, about the Grothendieck papers.)

## Archive for M-open inference

## La déraisonnable efficacité des mathématiques

Posted in Books, pictures, Statistics, University life with tags all models are wrong, Bayesian inference, foundations, France Culture, France Inter, Grothendieck, interview, M-open inference, Paris, philosophy of sciences, Radio France, statistical mathematics on May 11, 2017 by xi'an## a unified treatment of predictive model comparison

Posted in Books, Statistics, University life with tags AIC, Bayesian model comparison, Bayesian predictive, Bourbaki, DIC, Kullback-Leibler divergence, M-open inference, marginal likelihood, posterior predictive, small worlds on June 16, 2015 by xi'an

“Applying various approximation strategies to the relative predictive performance derivedfrom predictive distributions in frequentist and Bayesian inference yields many of the modelcomparison techniques ubiquitous in practice, from predictive log loss cross validation tothe Bayesian evidence and Bayesian information criteria.”

**M**ichael Betancourt (Warwick) just arXived a paper formalising predictive model comparison in an almost Bourbakian sense! Meaning that he adopts therein a very general representation of the issue, with minimal assumptions on the data generating process (excluding a specific metric and obviously the choice of a testing statistic). He opts for an M-open perspective, meaning that this generating process stands outside the hypothetical statistical model or, in Lindley’s terms, a small world. Within this paradigm, the only way to assess the fit of a model seems to be through the predictive performances of that model. Using for instance an f-divergence like the Kullback-Leibler divergence, based on the true generated process as the reference. I think this however puts a restriction on the choice of small worlds as the probability measure on that small world has to be absolutely continuous wrt the true data generating process for the distance to be finite. While there are arguments in favour of absolutely continuous small worlds, this assumes a knowledge about the true process that we simply cannot gather. Ignoring this difficulty, a relative Kullback-Leibler divergence can be defined in terms of an almost arbitrary reference measure. But as it still relies on the true measure, its evaluation proceeds via cross-validation “tricks” like jackknife and bootstrap. However, on the Bayesian side, using the prior predictive links the Kullback-Leibler divergence with the marginal likelihood. And Michael argues further that the posterior predictive can be seen as the unifying tool behind information criteria like DIC and WAIC (widely applicable information criterion). Which does not convince me towards the utility of those criteria as model selection tools, as there is too much freedom in the way approximations are used and a potential for using the data several times.