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!

The Search for Certainty

“The expected value is hardly ever expected.” The Search for Certainty, page 207.

Following the advice of an Og’s reader, Erkan, I read Krzysztof Burdzy’s The Search for Certainty over the past weeks (and mostly in the métro). The author is primarily a probabilist, but he has set upon launching a radical criticism of the philosophical foundations of Statistics, the book subtitle being (somehow misleadingly) put as On the Clash of Science and Philosophy of Probability. Or so he thinks. (The introduction to the book is available on the author’s webpage and he has also started a blog related with the book. The cover is made of seven dices all showing sixes, with on top the probability p=0.0000036… of such an event, provided the dice is fair. And provided one does not take into account the suspicious fives all facing the reader!)

“Of the four well crystallized philosophies of probability, two chose the certainty as their intellectual holy grail. Those are the failed theories of von Mises and de Finetti.” The Search for Certainty, page 30.

Although I have read the book with a pen, I will not go at this stage into a detailed analysis of The Search for Certainty. Indeed, I found the book both annoying and unconvincing, for reasons not very different from the criticisms addressed at Taleb’s The Black Swan. The book aims at demonstrating that the philosophical arguments underlying both frequentist and Bayesian Statistics are wrong. Unsurprisingly (!), I find the book lacking in this demonstration and overall poor from a scholarly perspective. It compares with Taleb’s The Black Swan in that the attempts at philosophy are more related to everyday “common sense” than to deep (and scholarly) philosophy (and they also involve the apparently inevitable Karl Popper, “the champion of the propensity theory of probability”, p.43!). The main point made in The Search for Certainty is very narrow in that Burdzy concentrates on two very specific entries to frequentism and subjectivism, namely von Mises’ and de Finetti’s, respectively, while those are not your average statistician’s references. For instance, von Mises bases his definition of frequency properties on the notion of collectives, a notion I had not previously encountered. Similarly, de Finetti’s statement “Probability does not exist” cannot be seen as the core principle of many Bayesian statisticians and I certainly do not relate to his all-subjective perspective for conducting Bayesian inference.

“Mr. Winston is unique because we know something about him that we do not know about any other individual in the population.” The Search for Certainty, page 66.

The style of The Search for Certainty is fairly annoying, if of a different sort than Taleb‘s. It is predominantly non-technical, the worked-out examples being always of the coin and balls-in-an-urn type. Arguments are never more than one paragraph long and metaphors and weak analogies are more confusing than helpful. The book criticises a lot decision-theory and the related coherence of Bayesian procedures, ie against the Dutch Book argument, but it misses the connections between frequentist optimality(ies) and Bayesian procedures, like Wald’s complete class theorem, Welch and Peers (1964) matching priors, or the more recent Berger’s frequentist-Bayesian perspective. There are also (in my opinion) confusions, as when the basis for the frequency approach to probability is criticised for being connected with an unrealistic infinity of events, thus confusing concepts with experiments (indeed, the Large Law of Large Numbers cannot be proved by an experiment), the existence of a model versus its assessment (“Kolmogorov’s axioms say nothing on how to match the mathematical results with reality”, p.31), the use of a probability against its “truth”, the inclusion of time (and thus model shifts) into mathematical axioms, the assimilation of frequentist statistics to unbiased estimators, a somehow diffuse belief that some priors can be proven to be better than others, an argument that  they can be evaluated by their predictive performances… The insistence in adding new axioms to Kolmogorov’s is furthermore puzzling:

(L4) If there exists a symmetry on the space of possible outcomes which maps an event A onto an event B then the two events have equal probabilities, that is, P(A) = P(B).
(L5) An event has probability 0 if and only if it cannot occur. An event has probability 1 if and only if it must occur.

The axiom (L4) relates to both the Principle of Insufficient Reasons, whose limitations are the cause for much debate in the selection of prior distributions, and to invariance principles that lead to Haar measures as default noninformative priors. But I do not see the point in adding such an axiom into the tenets of probability. (And (L5) is more psychological than mathematical…)

“There is no justification for the use of the Bayes theorem in the subjective theory.” The Search for Certainty, page 144.

In conclusion, it is hopefully obvious that I did not overly enjoy this The Search for Certainty and that I do not consider it makes a significant contribution to the foundations of statistical inference and in particular to Bayesian analysis. Being examined by an outsider to our discipline certainly has a strong appeal, but only if done at a deep enough level.

Ps-Be warned that there is a homonymous book with full titlde The Search for Certainty: A Philosophical Account of Foundations of Mathematics on issues related with Gödel’s incompleteness theorem, by Marcus Gianquinto, book that I mistakenly bought for The Search for Certainty discussed here. Professor Gianquinto is actually a professor of philosophy at UCL, who specialises in epistemology so he could also comment on this book.