Bayesian inference: challenges, perspectives, and prospects

Over the past year, Judith, Michael and I edited a special issue of Philosophical Transactions of the Royal Society on Bayesian inference: challenges, perspectives, and prospects, in celebration of the current President of the Royal Society, Adrian Smith, and his contributions to Bayesian analysis that have impacted the field up to this day. The issue is now out! The following is the beginning of our introduction of the series.

When contemplating his past achievements, it is striking to align the emergence of massive advances in these fields with some papers or books of his. For instance, Lindley’s & Smith’s ‘Bayes Estimates for the Linear Model’ (1971), a Read Paper at the Royal Statistical Society, is making the case for the Bayesian analysis of this most standard statistical model, as well as emphasizing the notion of exchangeability that is foundational in Bayesian statistics, and paving the way to the emergence of hierarchical Bayesian modelling. It thus makes a link between the early days of Bruno de Finetti, whose work Adrian Smith translated into English, and the current research in non-parametric and robust statistics. Bernardo’s & Smith’s masterpiece, Bayesian Theory (1994), sets statistical inference within decision- and information-theoretic frameworks in a most elegant and universal manner that could be deemed a Bourbaki volume for Bayesian statistics if this classification endeavour had reached further than pure mathematics. It also emphasizes the central role of hierarchical modelling in the construction of priors, as exemplified in Carlin’s et al.‘Hierarchical Bayesian analysis of change point problems’ (1992).

The series of papers published in 1990 by Alan Gelfand & Adrian Smith, esp. ‘Sampling-Based Approaches to Calculating Marginal Densities’ (1990), is overwhelmingly perceived as the birth date of modern Markov chain Monte Carlo (MCMC) methods, as itbrought to the whole statistics community (and the quickly wider communities) the realization that MCMC simulation was the sesame to unlock complex modelling issues. The consequences on the adoption of Bayesian modelling by non-specialists are enormous and long-lasting.Similarly, Gordon’set al.‘Novel approach to nonlinear/non-Gaussian Bayesian state estimation’ (1992) is considered as the birthplace of sequential Monte Carlo, aka particle filtering, with considerable consequences in tracking, robotics, econometrics and many other fields. Titterington’s, Smith’s & Makov’s reference book, ‘Statistical Analysis of Finite Mixtures’ (1984)  is a precursor in the formalization of heterogeneous data structures, paving the way for the incoming MCMC resolutions like Tanner & Wong (1987), Gelman & King (1990) and Diebolt & Robert (1990). Denison et al.’s book, ‘Bayesian methods for nonlinear classification and regression’ (2002) is another testimony to the influence of Adrian Smith on the field,stressing the emergence of robust and general classification and nonlinear regression methods to analyse complex data, prefiguring in a way the later emergence of machine-learning methods,with the additional Bayesian assessment of uncertainty. It is also bringing forward the capacity of operating Bayesian non-parametric modelling that is now broadly accepted, following a series of papers by Denison et al. in the late 1990s like CART and MARS.

We are quite grateful to the authors contributing to this volume, namely Joshua J. Bon, Adam Bretherton, Katie Buchhorn, Susanna Cramb, Christopher Drovandi, Conor Hassan, Adrianne L. Jenner, Helen J. Mayfield, James M. McGree, Kerrie Mengersen, Aiden Price, Robert Salomone, Edgar Santos-Fernandez, Julie Vercelloni and Xiaoyu Wang, Afonso S. Bandeira, Antoine Maillard, Richard Nickl and Sven Wang , Fan Li, Peng Ding and Fabrizia Mealli, Matthew Stephens, Peter D. Grünwald, Sumio Watanabe, Peter Müller, Noirrit K. Chandra and Abhra Sarkar, Kori Khan and Alicia Carriquiry, Arnaud Doucet, Eric Moulines and Achille Thin, Beatrice Franzolini, Andrea Cremaschi, Willem van den Boom and Maria De Iorio, Sandra Fortini and Sonia Petrone, Sylvia Frühwirth-Schnatter, Sara Wade, Chris C. Holmes and Stephen G. Walker, Lizhen Nie and Veronika Ročková. Some of the papers are open-access, if not all, hence enjoy them!

maison Poincaré

prime suspects [book review]

I was contacted by Princeton University Press to comment on the comic book/graphic novel Prime Suspects (The Anatomy of Integers and Permutations), by Andrew Granville (mathematician) & Jennifer Granville (writer), and Robert Lewis (illustrator), and they sent me the book. I am not a big fan of graphic book entries to mathematical even less than to statistical notions (Logicomix being sort of an exception for its historical perspective and nice drawing style) and this book did nothing to change my perspective on the subject. First, the plot is mostly a pretense at introducing number theory concepts and I found it hard to follow it for more than a few pages. The [noires maths] story is that “forensic maths” detectives are looking at murders that connects prime integers and permutations… The ensuing NCIS-style investigation gives the authors the opportunity to skim through the whole cenacle of number theorists, plus a few other mathematicians, who appear as more or less central characters. Even illusory ones like Nicolas Bourbaki. And Alexander Grothendieck as a recluse and clairvoyant hermit [who in real life did not live in a Pyrénées cavern!!!]. Second, I [and nor is Andrew who was in my office when the book arrived!] am not particularly enjoying the drawings or the page composition or the colours of this graphic novel, especially because I find the characters drawn quite inconsistently from one strip to the next, to the point of being unrecognisable, and, if it matters, hardly resembling their real-world equivalent (as seen in the portrait of Persi Diaconis). To be completely honest, the drawings look both ugly and very conventional to me, in that I do not find much of a characteristic style to them. To contemplate what Jacques Tardi,  François Schuiten or José Muñoz could have achieved with the same material… (Or even Edmond Baudoin, who drew the strips for the graphic novels he coauthored with Cédric Villani.) The graphic novel (with a prime 181 pages) is postfaced with explanations about the true persons behind the characters, from Carl Friedriech Gauß to Terry Tao, and of course on the mathematical theory for the analogies between the prime and cycles frequencies behind the story. Which I find much more interesting and readable, obviously. (With a surprise appearance of Kingman’s coalescent!) But also somewhat self-defeating in that so much has to be explained on the side for the links between the story, the characters and the background heavily loaded with “obscure references” to make sense to more than a few mathematician readers. Who may prove to be the core readership of this book.

There is also a bit of a Gödel-Escher-and-Bach flavour in that a piece by Robert Schneider called Réverie in Prime Time Signature is included, while an Escher’s infinite stairway appears in one page, not far from what looks like Milano Vittorio Emmanuelle gallery (On the side, I am puzzled by the footnote on p.208 that “I should clarify that selecting a random permutation and a random prime, as described, can be done easily, quickly, and correctly”. This may be connected to the fact that the description of Bach’s algorithm provided therein is incomplete.)

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]

death notice from Bourbaki