**Y**et another question on X validated reminded me of a discussion I had once with Jay Kadane when visiting Carnegie Mellon in Pittsburgh. Namely the fundamentally ill-posed nature of conjugate priors. Indeed, when considering the definition of a conjugate family as being a parameterised family Þ of distributions over the parameter space Θ stable under transform to the posterior distribution, this property is completely dependent (if there is such a notion as completely dependent!) on the dominating measure adopted on the parameter space Θ. Adopted is the word as there is no default, reference, natural, &tc. measure that promotes one specific measure on Θ as being *the* dominating measure. This is a well-known difficulty that also sticks out in most “objective Bayes” problems, as well as with maximum entropy priors. This means for instance that, while the Gamma distributions constitute a conjugate family for a Poisson likelihood, so do the truncated Gamma distributions. And so do the distributions which density (against a Lebesgue measure over an arbitrary subset of (0,∞)) is the product of a Gamma density by an arbitrary function of θ. I readily acknowledge that the standard conjugate priors as introduced in every Bayesian textbook are standard because they facilitate (to a certain extent) posterior computations. But, just like there exist an infinity of MaxEnt priors associated with an infinity of dominating measures, there exist an infinity of conjugate families, once more associated with an infinity of dominating measures. And the fundamental reason is that the sampling model (which induces the shape of the conjugate family) does not provide a measure on the parameter space Θ.

## Archive for Bayesian textbook

## dominating measure

Posted in Books, pictures, Statistics, Travel, University life with tags Bayesian textbook, Carnegie Mellon University, conjugate priors, cross validated, dominating measure, Jay Kadane, Pittsburgh, posterior distribution on March 21, 2019 by xi'an## I’m getting the point

Posted in Statistics with tags Bayesian statistics, Bayesian textbook, conjugate priors, cross validated, final exam, StackExchange, teaching on February 14, 2019 by xi'an**A** long-winded X validated discussion on the [textbook] mean-variance conjugate posterior for the Normal model left me [mildly] depressed at the point and use of answering questions on this forum. Especially as it came at the same time as a catastrophic outcome for my mathematical statistics exam. Possibly an incentive to quit X validated as one quits smoking, although this is not the first attempt…

## Computational Bayesian Statistics [book review]

Posted in Books, Statistics with tags ABC, Bayes factor, Bayesian model selection, Bayesian p-values, Bayesian paradigm, Bayesian textbook, BayesX, book review, Cambridge University Press, coda, computational Bayesian methods, cup, Gibbs sampling, information criterion, INLA, JAGS, Jeffreys prior, Kalman filter, Laplace approximation, Likelihood Principle, MCMC, Metropolis-Hastings algorithm, model assessment, Monte Carlo Statistical Methods, OpenBUGS, R, sequential Monte Carlo, STAN, subjective versus objective Bayes on February 1, 2019 by xi'an**T**his Cambridge University Press book by M. Antónia Amaral Turkman, Carlos Daniel Paulino, and Peter Müller is an enlarged translation of a set of lecture notes in Portuguese. *(Warning: I have known Peter Müller from his PhD years in Purdue University and cannot pretend to perfect objectivity. For one thing, Peter once brought me frozen-solid beer: revenge can also be served cold!)* Which reminds me of my 1994 French edition of Méthodes de Monte Carlo par chaînes de Markov, considerably upgraded into Monte Carlo Statistical Methods (1998) thanks to the input of George Casella. *(Re-warning: As an author of books on the same topic(s), I can even less pretend to objectivity.)*

“The “great idea” behind the development of computational Bayesian statistics is the recognition that Bayesian inference can be implemented by way of simulation from the posterior distribution.”

The book is written from a strong, almost militant, subjective Bayesian perspective (as, e.g., when *half-Bayesians* are mentioned!). Subjective (and militant) as in Dennis Lindley‘s writings, eminently quoted therein. As well as in Tony O’Hagan‘s. Arguing that the sole notion of a Bayesian estimator is the entire posterior distribution. Unless one brings in a loss function. The book also discusses the Bayes factor in a critical manner, which is fine from my perspective. (Although the ban on improper priors makes its appearance in a very indirect way at the end of the last exercise of the first chapter.)

Somewhat at odds with the subjectivist stance of the previous chapter, the chapter on prior construction only considers non-informative and conjugate priors. Which, while understandable in an introductory book, is a wee bit disappointing. (When mentioning Jeffreys’ prior in multidimensional settings, the authors allude to using univariate Jeffreys’ rules for the marginal prior distributions, which is not a well-defined concept or else Bernardo’s and Berger’s reference priors would not have been considered.) The chapter also mentions the likelihood principle at the end of the last exercise, without a mention of the debate about its derivation by Birnbaum. Or Deborah Mayo’s recent reassessment of the strong likelihood principle. The following chapter is a sequence of illustrations in classical exponential family models, classical in that it is found in many Bayesian textbooks. (Except for the Poison model found in Exercise 3.3!)

Nothing to complain (!) about the introduction of Monte Carlo methods in the next chapter, especially about the notion of inference by Monte Carlo methods. And the illustration by Bayesian design. The chapter also introduces Rao-Blackwellisation [prior to introducing Gibbs sampling!]. And the simplest form of bridge sampling. (Resuscitating the weighted bootstrap of Gelfand and Smith (1990) may not be particularly urgent for an introduction to the topic.) There is furthermore a section on sequential Monte Carlo, including the Kalman filter and particle filters, in the spirit of Pitt and Shephard (1999). This chapter is thus rather ambitious in the amount of material covered with a mere 25 pages. Consensus Monte Carlo is even mentioned in the exercise section.

“This and other aspects that could be criticized should not prevent one from using this [Bayes factor] method in some contexts, with due caution.”

Chapter 5 turns back to inference with model assessment. Using Bayesian p-values for model assessment. (With an harmonic mean spotted in Example 5.1!, with no warning about the risks, except later in 5.3.2.) And model comparison. Presenting the whole collection of xIC information criteria. from AIC to WAIC, including a criticism of DIC. The chapter feels somewhat inconclusive but methinks this is the right feeling on the current state of the methodology for running inference about the model itself.

“Hint: There is a very easy answer.”

Chapter 6 is also a mostly standard introduction to Metropolis-Hastings algorithms and the Gibbs sampler. (The argument given later of a Metropolis-Hastings algorithm with acceptance probability one does not work.) The Gibbs section also mentions demarginalization as a [latent or auxiliary variable] way to simulate from complex distributions [as we do], but without defining the notion. It also references the precursor paper of Tanner and Wong (1987). The chapter further covers slice sampling and Hamiltonian Monte Carlo, the later with sufficient details to lead to reproducible implementations. Followed by another standard section on convergence assessment, returning to the 1990’s feud of single versus multiple chain(s). The exercise section gets much larger than in earlier chapters with several pages dedicated to most problems. Including one on ABC, maybe not very helpful in this context!

“…dimension padding (…) is essentially all that is to be said about the reversible jump. The rest are details.”

The next chapter is (somewhat logically) the follow-up for trans-dimensional problems and marginal likelihood approximations. Including Chib’s (1995) method [with no warning about potential biases], the spike & slab approach of George and McCulloch (1993) that I remember reading in a café at the University of Wyoming!, the somewhat antiquated MC³ of Madigan and York (1995). And then the much more recent array of Bayesian lasso techniques. The trans-dimensional issues are covered by the pseudo-priors of Carlin and Chib (1995) and the reversible jump MCMC approach of Green (1995), the later being much more widely employed in the literature, albeit difficult to tune [and even to comprehensively describe, as shown by the algorithmic representation in the book] and only recommended for a large number of models under comparison. Once again the exercise section is most detailed, with recent entries like the EM-like variable selection algorithm of Ročková and George (2014).

The book also includes a chapter on analytical approximations, which is also the case in ours [with George Casella] despite my reluctance to bring them next to exact (simulation) methods. The central object is the INLA methodology of Rue et al. (2009) [absent from our book for obvious calendar reasons, although Laplace and saddlepoint approximations are found there as well]. With a reasonable amount of details, although stopping short of implementable reproducibility. Variational Bayes also makes an appearance, mostly following the very recent Blei et al. (2017).

The gem and originality of the book are primarily to be found in the final and ninth chapter where four software are described, all with interfaces to R: OpenBUGS, JAGS, BayesX, and Stan, plus R-INLA which is processed in the second half of the chapter (because this is *not* a simulation method). As in the remainder of the book, the illustrations are related to medical applications. Worth mentioning is the reminder that BUGS came in parallel with Gelfand and Smith (1990) Gibbs sampler rather than as a consequence. Even though the formalisation of the Markov chain Monte Carlo principle by the later helped in boosting the power of this software. (I also appreciated the mention made of Sylvia Richardson’s role in this story.) Since every software is illustrated in depth with relevant code and output, and even with the shortest possible description of its principle and *modus vivendi*, the chapter is 60 pages long [and missing a comparative conclusion]. Given my total ignorance of the very existence of the BayesX software, I am wondering at the relevance of its inclusion in this description rather than, say, other general R packages developed by authors of books such as Peter Rossi. The chapter also includes a description of CODA, with an R version developed by Martin Plummer [now a Warwick colleague].

In conclusion, this is a high-quality and all-inclusive introduction to Bayesian statistics and its computational aspects. By comparison, I find it much more ambitious and informative than Albert’s. If somehow less pedagogical than the thicker book of Richard McElreath. (The repeated references to Paulino et al. (2018) in the text do not strike me as particularly useful given that this other book is written in Portuguese. Unless an English translation is in preparation.)

*Disclaimer: this book was sent to me by CUP for endorsement and here is what I wrote in reply for a back-cover entry:*

An introduction to computational Bayesian statistics cooked to perfection, with the right mix of ingredients, from the spirited defense of the Bayesian approach, to the description of the tools of the Bayesian trade, to a definitely broad and very much up-to-date presentation of Monte Carlo and Laplace approximation methods, to an helpful description of the most common software. And spiced up with critical perspectives on some common practices and an healthy focus on model assessment and model selection. Highly recommended on the menu of Bayesian textbooks!

And this review is likely to appear in CHANCE, in my book reviews column.

## a glaring mistake

Posted in Statistics with tags Bayes factor, Bayesian hypothesis testing, Bayesian textbook, cross validated, Stack Exchange, The Bayesian Choice, typos on November 28, 2018 by xi'an**S**omeone posted this question about Bayes factors in my book on Saturday morning and I could not believe the glaring typo pointed out there had gone through the centuries without anyone noticing! There should be no index 0 or 1 on the θ’s in either integral (or indices all over). I presume I made this typo when cutting & pasting from the previous formula (which addressed the case of two point null hypotheses), but I am quite chagrined that I sabotaged the definition of the Bayes factor for generations of readers of the Bayesian Choice. Apologies!!!

## ghost [parameters] in the [Bayesian] shell

Posted in Books, Kids, Statistics with tags Bayesian model comparison, Bayesian textbook, Brad Carlin, cross validated, Doing Bayesian Data Analysis, model posterior probabilities, Sid Chib, Stack Exchange on August 3, 2017 by xi'an**T**his question appeared on Stack Exchange (X Validated) two days ago. And the equalities indeed seem to suffer from several mathematical inconsistencies, as I pointed out in my Answer. However, what I find most crucial in this question is that the quantity on the left hand side is meaningless. Parameters for different models only make sense within their own model. Hence when comparing models parameters cannot co-exist across models. What I suspect [without direct access to Kruschke’s Doing Bayesian Data Analysis book and as was later confirmed by John] is that he is using pseudo-priors in order to apply Carlin and Chib (1995) resolution [by saturation of the parameter space] of simulating over a trans-dimensional space…

## Peter Lee (1940?-2017)

Posted in Books, pictures, R, Statistics, University life, Wines with tags Bayesian statisticians, Bayesian textbook, England, obituary, Peter Lee, R, York on March 12, 2017 by xi'an**J**ust heard the sad news that Peter Lee, British Bayesian and author of Bayesian Statistics: An Introduction, has passed away yesterday night. While I did not know him, I remember meeting him at a few conferences in the UK and spending an hilarious evening at the pub. When the book came out, I thought it was quite fine an introduction to Bayesian Statistics, with enough mathematical details and prerequisites to make it worthwhile studying, while also including computational recommendations. Fare thee well, Peter.