Bayesian restricted likelihood with insufficient statistic [slides]

A great Bayesian Analysis webinar this afternoon with well-balanced presentations by Steve MacEachern and John Lewis, and original discussions by Bertrand Clarke and Fabrizio Rugieri. Which attracted 122 participants. I particularly enjoyed Bertrand’s points that likelihoods were more general than models [made in 6 different wordings!] and that this paper was closer to the M-open perspective. I think I eventually got the reason why the approach could be seen as an ABC with ε=0, since the simulated y’s all get the right statistic, but this presentation does not bring a strong argument in favour of the restricted likelihood approach, when considering the methodological and computational effort. The discussion also made me wonder if tools like VAEs could be used towards approximating the distribution of T(y) conditional on the parameter θ. This is also an opportunity to thank my friend Michele Guindani for his hard work as Editor of Bayesian Analysis and in particular for keeping the discussion tradition thriving!

La déraisonnable efficacité des mathématiques

Although 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.)

a unified treatment of predictive model comparison

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

Michael 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.