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!

optimal estimation of parameters (book review)

As I had read some of Jorma Rissanen’s papers in the early 1990’s when writing The Bayesian Choice, I was quite excited to learn that Rissanen had written a book on the optimal estimation of parameters, where he presents and develops his own approach to statistical inference (estimation and testing). As explained in the Preface this was induced by having to deliver the 2009 Shannon Lecture at the Information Theory Society conference.

“Very few statisticians have been studying information theory, the result of which, I think, is the disarray of the present discipline of statistics.” J. Rissanen (p.2)

Now that I have read the book (between Venezia in the peaceful and shaded Fundamenta Sacca San Girolamo and Hong Kong, so maybe in too a leisurely and off-handed manner), I am not so excited… It is not that the theory presented in optimal estimation of parameters is incomplete or ill-presented: the book is very well-written and well-designed, if in a highly personal (and borderline lone ranger) style. But the approach Rissanen advocates, namely maximum capacity as a generalisation of maximum likelihood, does not seem to relate to my statistical perspective and practice. Even though he takes great care to distance himself from Bayesian theory by repeating that the prior distribution is not necessary for his theory of optimal estimation (“priors are not needed in the general MDL principle”, p.4). my major source of incomprehension lies with the choice of incorporating the estimator within the data density to produce a new density, as in

Indeed, this leads to (a) replace a statistical model with a structure that mixes the model and the estimation procedure and (b) peak the new distribution by always choosing the most appropriate (local) value of the parameter. For a normal sample with unknown mean θ, this produces for instance to a joint normal distribution that is degenerate since

(For a single observation it is not even defined.) In a similar spirit, Rissanen defines this estimated model for dynamic data in a sequential manner, which means in the end that x₁ is used n times, x₂ n-1 times, and so on.., This asymmetry does not sound logical, especially when considering sufficiency.

“…the misunderstanding that the more parameters there are in the model the better it is because it is closer to the `truth’ and the `truth’ obviously is not simple” J. Rissanen (p.38)

Another point of contention with the approach advocated in optimal estimation of parameters is the inherent discretisation of the parameter space, which seems to exclude large dimensional spaces and complex models. I somehow subscribe to the idea that a given sample (hence a given sample size) induces a maximum precision in the estimation that can be translated into using a finite number of parameter values, but the implementation suggested in the book is essentially unidimensional. I also find the notion of optimality inherent to the statistical part of optimal estimation of parameters quite tautological as it ends up being a target that leads to the maximum likelihood estimator (or its pseudo-Bayesian counterpart).

“The BIC criterion has neither information nor a probability theoretic interpretation, and it does not matter which measure for consistency is selected.” J. Rissanen (p.64)

The first part of the book is about coding and information theory; it amounts in my understanding to a justification of the Kullback-Leibler divergence, with an early occurrence (p.27) of the above estimation distribution. (The channel capacity is the normalising constant of this weird density.)

“…in hypothesis testing [where] the assumptions that the hypotheses are  `true’ has misguided the entire field by generating problems which do not exist and distorting rational solutions to problems that do exist.” J. Rissanen (p.41)

I have issues with the definition of confidence intervals as they rely on an implicit choice of a measure and have a constant coverage that decreases with the parameter dimension. This notion also seem to clash with the subsequent discretisation of the parameter space. Hypothesis testing à la Rissanen reduces to an assessment of a goodness of fit, again with fixed coverage properties. Interestingly, the acceptance and rejection regions are based on two quantities, the likelihood ratio and the KL distance (p. 96), which leads to a delayed decision if they do not agree wrt fixed bounds.

“A drawback of the prediction formulas is that they require the knowledge of the ARMA parameters.” J. Rissanen (p.141)

A final chapter on sequential (or dynamic) models reminded me that Rissanen was at the core of inventing variable order Markov chains. The remainder of this chapter provides some properties of the sequential normalised maximum likelihood estimator advocated by the author in the same spirit as the earlier versions.  The whole chapter feels (to me) somewhat disconnected from

In conclusion, Rissanen’s book is a definitely  interesting  entry on a perplexing vision of statistics. While I do not think it will radically alter our understanding and practice of statistics, it is worth perusing, if only to appreciate there are still people (far?) out there attempting to bring a new vision of the field.

mathematical statistics books with Bayesian chapters [incomplete book reviews]

I received (in the same box) two mathematical statistics books from CRC Press, Understanding Advanced Statistical Methods by Westfall and Henning, and Statistical Theory A Concise Introduction by Abramovich and Ritov. For review in CHANCE. While they are both decent books for teaching mathematical statistics at undergraduate borderline graduate level, I do not find enough of a novelty in them to proceed to a full review. (Given more time, I could have changed my mind about the first one.) Instead, I concentrate here on their processing of the Bayesian paradigm, which takes a wee bit more than a chapter in either of them. (And this can be done over a single métro trip!) The important following disclaimer applies: comparing both books is highly unfair in that it is only because I received them together. They do not necessarily aim at the same audience. And I did not read the whole of either of them.

First, the concise Statistical Theory  covers the topic in a fairly traditional way. It starts with a warning about the philosophical nature of priors and posteriors, which reflect beliefs rather than frequency limits (just like likelihoods, no?!). It then introduces priors with the criticism that priors are difficult to build and assess. The two classes of priors analysed in this chapter are unsurprisingly conjugate priors (which hyperparameters have to be determined or chosen or estimated in the empirical Bayes heresy [my words!, not the authors’]) and “noninformative (objective) priors”.  The criticism of the flat priors is also traditional and leads to the  group invariant (Haar) measures, then to Jeffreys non-informative priors (with the apparent belief that Jeffreys only handled the univariate case). Point estimation is reduced to posterior expectations, confidence intervals to HPD regions, and testing to posterior probability ratios (with a warning about improper priors). Bayes rules make a reappearance in the following decision-theory chapter, as providers of both admissible and minimax estimators. This is it, as Bayesian techniques are not mentioned in the final “Linear Models” chapter. As a newcomer to statistics, I think I would be as bemused about Bayesian statistics as when I got my 15mn entry as a student, because here was a method that seemed to have a load of history, an inner coherence, and it was mentioned as an oddity in an otherwise purely non-Bayesian course. What good could this do to the understanding of the students?! So I would advise against getting this “token Bayesian” chapter in the book…

“You are not ignorant! Prior information is what you know prior to collecting the data.” Understanding Advanced Statistical Methods (p.345)

Second, Understanding Advanced Statistical Methods offers a more intuitive entry, by justifying prior distributions as summaries of prior information. And observations as a mean to increase your knowledge about the parameter. The Bayesian chapter uses a toy but very clear survey examplew to illustrate the passage from prior to posterior distributions. And to discuss the distinction between informative and noninformative priors. (I like the “Ugly Rule of Thumb” insert, as it gives a guideline without getting too comfy about it… E.g., using a 90% credible interval is good enough on p.354.) Conjugate priors are mentioned as a result of past computational limitations and simulation is hailed as a highly natural tool for analysing posterior distributions. Yay! A small section discusses the purpose of vague priors without getting much into details and suggests to avoid improper priors by using “distributions with extremely large variance”, a concept we dismissed in Bayesian Core! For how large is “extremely large”?!

“You may end up being surprised to learn in later chapters (..) that, with classical methods, you simply cannot perform the types of analyses shown in this section (…) And that’s the answer to the question, “What good is Bayes?””Understanding Advanced Statistical Methods (p.345)

Then comes the really appreciable part, a section entitled “What good is Bayes?”—it actually reads “What Good is Bayes?” (p.359), leading to a private if grammatically poor joke since I.J. Good was one of the first modern Bayesians, working with Turing at Bletchley Park…—  The authors simply skip the philosophical arguments to give the reader a showcase of examples where the wealth of the Bayesian toolbox: logistic regression, VaR (Value at Risk), stock prices, drug profit prediction. Concluding with arguments in favour of the frequentist methods: (a) not requiring priors, (b) easier with generic distributifrequentistons, (c) easier to understand with simulation, and (d) easier to validate with validation. I do not mean to get into a debate about those points as my own point is that the authors are taking a certain stand about the pros and cons of the frequentist/Bayesian approaches and that they are making their readers aware of it. (Note that the Bayesian chapter comes before the frequentist chapter!) A further section is “Comparing the Bayesian and frequentist paradigms?” (p.384), again with a certain frequentist slant, but again making the distinctions and similarities quite clear to the reader. Of course, there is very little (if any) about Bayesian approaches in the next chapters but this is somehow coherent with the authors’ perspective. Once more, a perspective that is well-spelled and comprehensible for the reader. Even the novice statistician. In that sense, having a Bayesian chapter inside a general theory book makes sense.  (The second book has a rather detailed website, by the way! Even though handling simulations in Excel and drawing graphs in SAS could be dangerous to your health…)