Archive for objective Bayes
objective Bayesian inference now in paperback
Posted in Books, Statistics with tags advertising, amazon associates, Bayesian foundations, Bayesian inference, Book, Dongchu Sun, JASA, Jim Berger, José Miguel Bernardo, noninformative priors, objective Bayes, paperback, publication, reference priors, technology on May 1, 2026 by xi'an
model uncertainty and missing data: an objective BAyesian perspective
Posted in Books, Statistics, Travel, University life with tags Arnold Zellner, Bayes factor, Bayesian Analysis, Bayesian model comparison, Bayesian predictive, Bayesian variable selection, deviance information criterion, DIC, Don Rubin, Florence Forbes, g-priors, Gilles Celeux, Haar measure, ignorability, improper priors, invariance, Jeffreys priors, linear regression, marginal likelihood, Mike Titterington, missing values, missing-at-random model, model uncertainty, objective Bayes, objective prior distribution, oracle, ozone dataset, reference prior, Rubin’s rules, Spain, time zones, webinar on September 16, 2025 by xi'an
My Spanish and objective Bayesian friends Gonzalo García-Donato, María Eugenia Castellanos, Stefano Cabras, Alicia Quirós, and Anabel Forte wrote an fairly exciting paper in BA that is open to discussion (for a few more days), to be discussed on 05 November (4:00 PM UTC | 11:00 AM EST | 5:00 PM CET).
The interplay between missing data and model uncertainty—two classic statistical problems—leads to primary questions that we formally address from an objective Bayesian perspective. For the general regression problem, we discuss the probabilistic justification of Rubin’s rules applied to the usual components of Bayesian variable selection, arguing that prior predictive marginals should be central to the pursued methodology. In the regression settings, we explore the conditions of prior distributions that make the missing data mechanism ignorable, provided that it is missing at random or completely at random. Moreover, when comparing multiple linear models, we provide a complete methodology for dealing with special cases, such as variable selection or uncertainty regarding model errors. In numerous simulation experiments, we demonstrate that our method outperforms or equals others, in consistently producing results close to those obtained using the full dataset. In general, the difference increases with the percentage of missing data and the correlation between the variables used for imputation.
The so-called Rubin’s identity is simply the representation of the posterior probability of a model γ given the observed data x⁰, p(γ|x⁰), as the integrated posterior probability of a model given both observed and latent data, p(γ|x⁰, x¹), against the marginal of latent x¹ given observed x⁰. Since this marginal involves the probabilities p(γ|x⁰), this representation is not directly useful for a numerical implementation.
In this paper, missingness relates to some entries of either the covariates or the response variate. Which is less common but more realistic, especially if some covariates do not contribute to the response. (The missingness mechanism does not matter if the data is missing at random (à la Rubin). The computational solution (p9) is rather standard, simulating the missing variables given the observed variables. In my opinion, the elephant in the room is the super-delicate selection of a prior distribution on the missing covariates, as methinks this impacts in a considerable manner the actual value of the Bayes factor, hence the selection of the surviving model. (As a side remark, we are credited in Celeux et al. (2006) to have “extended DIC for missing data models or when missing data were present”, but our point was instead to point out the arbitrariness of the very definition of DIC in such contexts.)
“The standard Bayesian method for addressing the absence of prior information uses improper distributions. In estimation problems (the model is fixed), the impropriety of priors does not imply any additional difficulty as long as the posterior is proper” (p9)
The authors point out the well-known difficulty with improper priors but still resort to improper priors on the parameters shared by all models—which I dispute as being adequate, despite the arguments put forward on p15, right Haar measure or not—, while sticking to proper priors on the model-dependent parameters. Which unsurprisingly become Zellner’s g-priors. Or rather g’-priors, although the discussion seems to resolve into the (model-free) factor g’ being equal to 1 as for the g-priors. Again a strong term in the derivation of the Bayes factor.
O’Bayes 2025 [με αθηναϊκό άρωμα]
Posted in Statistics with tags Athens, Athens University of Economics and Business, BayesComp 2025, conference, foundations of objective Bayesian methodology, Greece, ISBA, ISBA conference, National Technical University of Athens, National University Singapore, O'Bayes, O'Bayes 2022, O'Bayes 2025, objective Bayes, Parthenon, Renzo Piano, scientific committee, Singapore, Stavros Niarchos Foundation, tutorial on November 6, 2024 by xi'an
The next O’Bayes conference will take place in Athens, 8-12 June 2025. Registration is open, as well as a call for posters. As in (all?) earlier episodes of these Objective Bayes methodology conferences, speakers and discussants are invited by the scientific committee. And the first afternoon consists of tutorials on different aspects of Objective Bayes methodology. The conference will take place in the Stavros Niarchos Foundation Cultural Centre. (Unfortunately, I will not be able to attend O’Bayes 2025, once again, due to my attending BayesComp 2025 the week after.)
Objective Bayesian Inference [book review]
Posted in Books, Statistics, University life with tags Bayesian Analysis, Bayesian textbook, book review, CHANCE, conjugate priors, Dongchu Sun, exponential families, foundations of objective Bayesian methodology, Harold Jeffreys, improper posteriors, insufficient reason, Jeffreys priors, Jim Berger, José Miguel Bernardo, maximum entropy, mixtures of distributions, non-informative priors, objective Bayes, Pierre Simon Laplace, publication, reference priors, Thomas Bayes' portrait on July 2, 2024 by xi'an
As advertised earlier on the ‘Og, the reference book on reference priors and relatives by my long-time friends Jim Berger, José Bernardo, and Dongchu Sun is at last out! I received a copy from the editor, World Scientific, and read through it, mostly in train rides to Normandy and Brittany. The construction of this book took decades and I remember many O’Bayes meetings when we were discussing of the progress made that far. As I knew from a few months back that the book was at last completed, I was quite eager to dig into it. And get this review ready for ISBA 2024. Given this prior knowledge, completed with sequential observations, I thus fear my review will be far from objective! And most likely more critical than it should be as fantasying how I would have written a book on that topic…
“Some of the best statisticians (not named Fisher or Neyman)…” (p1)
The book covers traditional approaches to principled ways of selecting prior distributions, culminating with the reference prior introduced by José Bernardo in his PhD thesis in the late 1970’s and expanded by all three authors over their academic careers. (Why is the acute accent missing from José on the front pages?!) The cover connects to the three founding fathers of objective Bayesian inference, Bayes, Laplace, and Jeffreys. The contents are not overly surprising from a personal viewpoint, i.e. as a card-carrying O’Bayes member. Namely that the chapters set the scene of parametric models and Bayesian inference (“a data driven probability transformation machine”), mostly supported by decision theory (including intrinsic losses!) but skipping testing and (mostly) model choice. This is unsurprisingly in the same spirit as Berger (1985) and Bernardo & Smith (1992). Not covering advanced Bayesian asymptotics, any flavour of Bayesian nonparametrics, the more recent generalized Bayesian inference, and the impact of misspecified models. The likelihood section does not mention Deborah Mayo’s criticism of the Likelihood Principle, or the Pitman Koopman lemma (although the examples are predominantly connected with exponential families). The section (1.8) on MCMC implies that the Metropolis algorithm is less accurate that the Gibbs sampler, which is an exaggerated generalisation from a simple example, accrued by a comparison that does not seem to account for mixing behaviours.
“Our own belief is that the effort [seeking objective prior distributions] is a misguided search for the holy grail” (p.68)
The basics of objective priors repeats the useful warning that a truncation of parameter space is far from advised, as is the call for vague proper priors à la BUGS (a “nonsense”). A remark on the alternative weakly informative priors à la BDA require subjective input, a whole section on the legitimacy of improper priors as KL limits of sequences of proper priors. Plus a nice recall of the data dependent prior of Wasserman (2000) forcing mixtures to avoid empty clusters. This was the prior Jean Diebolt and I implemented in our 1990 Gibbs sampling paper. (I do not really see it as data dependent to impose that no component comes empty in the sample, but rather as a different model removing some terms from the likelihood.) There is even a chapter dedicated to constant priors—the historical meaning of inverse probability—, with a section on the modern advocates of this constant prior, that mostly focus on Binomial model. The book goes on justifying this prior by an invariance under reparameterisation argument (p108), but the discussion may seem stretched for some newcomers. This is followed by a nice chapter on frequentist matching, covering the bivariate Normal case and some asymptotics, followed by confidence distributions, quite topical and then fiducial inference, that imagines a posterior without a prior, one short too short chapter on invariance priors arguing for the right-Haar vs the left-Haar prior measure in invariance settings as exact matching, completed by a useful if short chapter—I would not have thought of including—on the performances of objective priors, like over-dispersion or under-dispersion. A mention is made there of the (now well-known) danger of using MCMC with improper posteriors as the issue potentially goes undetected, as it did in the early 1990s. Within its coherence section, the authors recall the fundamentals of the lovely marginalisation paradoxes. While addressing some computational issues, the book does not mention the derivation of the Jeffreys prior for mixtures Clara Grazian and I examined. The chapter ends by a rather expedited dismissal of maximum entropy. Which many still regard as the default approach to (partly informed) objective prior modelling. (They’ll be back in Chapter 13.)
The last hundred pages (Chap. 9-14) of the book are focussing on reference priors, as should be given the priorities of the authors. Starting with the rather convincing concept of maximising missing information, getting asymptotic to remove the impact of the data, and turning recursive in case of multidimensional parameters, while resorting to compact parameter spaces to avoid improprieties (the Achille’s heel of reference priors!). Reaching a definition (p176) in the univariate case that coherently does not depend on the sample size but on an arbitrary dominating measure (p179), interestingly sharing this feature with the definition of conjugate priors. In multivariate settings, things get… more complicated! And force a separation between nuisance and interesting parameters lest the resulting priors prove underperforming. Asymptotic normality again helps, but the derivation remains involved witness a one page (p201) theorem (Proposition 10.2). My favourite example of selecting the prior for a Normal mean squared norm is there, with the original Jeffreys prior based on the Normal vector failing badly while the reference prior based on the norm of the observation does much better! (An open problem is the construction of the Jeffreys prior in that example.) A large table (p210) illustrates the plethora of reference priors depending on the parameter ordering. A short chapter (11) specialises on discrete parameters as in population sizes. And in model choice, a resolution I had not seen previously, with prior weights depending on the number of parameters in the respective models. But not accounting for embedded models. Chapter 12 addresses the “overall objective” prior construction when all parameters are equal (and none more equal than others). Supporting in the end the best overall prior defined in terms of distance to a family of reference priors. With a special treatment of the hierarchical Normal model following Berger & al. (2020). Chapter 13 is a short incursion into partial information reference priors, incl. maxent priors. Chapter 14 is about special reference priors exploiting special structures. And, at last, Chapter 15 a non-chapter pointing out to a catalogue of objective priors, following Yang & Berger (1997) as well as an initiative set during one of the O’Bayes meetings.
On the minor (nitpicking) side, I found a few “the the” (the typo no one can escape!) throughout the book, informality in some statements like Proposition 1.6, whose limit (in n) depends on n (a shortcut from which we try to wean our students). Also a somewhat anecdotal appearance of the ratio of uniforms algorithm with a mistaken statement that the method doesn’t depend on a proposal (p61), the references to Jeffreys’ main book clashing between the 1930s and 1961 (final edition). The “random posterior” section 1.8 6 seems unfinished.
In conclusion, this much awaited reference book does deliver! It brings a perspective on reference priors that no other book does and reflects (well) on the authors’ careful completion of a coherent theory, hence should appeal to anyone working on the foundations and principles of Bayesian inference. Obviously, it will not change the position of strict subjectivists, nor convince non-Bayesians, but it should inspire current and future researchers, as well as complement graduate courses on Bayesian inference. In addition, the huge bibliography retraces the work in the area till today (if less intensely in the most recent years). Kudos to the authors, then!
[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE!]
it’s objectively out!
Posted in Books, Statistics, University life with tags Bayesian Analysis, Bayesian textbook, Dongchu Sun, foundations of objective Bayesian methodology, Harold Jeffreys, insufficient reason, Jeffreys priors, Jim Berger, José Miguel Bernardo, non-informative priors, objective Bayes, Pierre Simon Laplace, publication, reference priors, Thomas Bayes' portrait on June 13, 2024 by xi'an
