Easy computation of the Bayes Factor

Posted in Books, Statistics with tags , , , , , on August 21, 2021 by xi'an

“Choosing the ranges has been criticized as introducing subjectivity; however, the key point is that the ranges are given quantitatively and should be justified”

On arXiv, I came across a paper by physicists Dunstan, Crowne, and Drew, on computing the Bayes factor by linear regression. Paper that I found rather hard to read given that the method is never completely spelled out but rather described through some examples (or the captions of figures)… The magical formula (for the marginal likelihood)

$B=(2\pi)^{n/2}L_{\max}\dfrac{\text{Cov}_p}{\prod_{i=1}^n \Delta p_i}$

where n is the parameter dimension, Cov is the Fisher information matrix, and the denominator the volume of a flat prior on an hypercube (!), seems to come for a Laplace approximation. But it depends rather crucially (!) on the choice of this volume. A severe drawback the authors evacuate with the above quote… And by using an example where the parameters have a similar meaning under both models. The following ones compare several dimensions of parameters without justifying (enough) the support of the corresponding priors. In addition, using a flat prior over the hypercube seems to clash with the existence of a (Fisher) correlation between the components. (To be completely open as to why I discuss this paper, I was asked to review the paper, which I declined.)

Computational Bayesian Statistics [book review]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , on February 1, 2019 by xi'an

This 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.

19 dubious ways to compute the marginal likelihood

Posted in Books, Statistics with tags , , , , , , , , , , on December 11, 2018 by xi'an

A recent arXival on nineteen different [and not necessarily dubious!] ways to approximate the marginal likelihood of a given topology of a philogeny tree that reminded me of our San Antonio survey with Jean-Michel Marin. This includes a version of the Laplace approximation called Laplus (!), accounting for the fact that branch lengths on the tree are positive but may have a MAP at zero. Using a Beta, Gamma, or log-Normal distribution instead of a Normal. For importance sampling, the proposals are derived from either the Laplus (!) approximate distributions or from the variational Bayes solution (based on an Normal product). Harmonic means are still used here despite the obvious danger, along with a defensive version that mixes prior and posterior. Naïve Monte Carlo means simulating from the prior, while bridge sampling seems to use samples from prior and posterior distributions. Path and modified path sampling versions are those proposed in 2008 by Nial Friel and Tony Pettitt (QUT). Stepping stone sampling appears like another version of path sampling, also based on a telescopic product of ratios of normalising constants, the generalised version relying on a normalising reference distribution that need be calibrated. CPO and PPD in the above table are two versions based on posterior predictive density estimates.

When running the comparison between so many contenders, the ground truth is selected as the values returned by MrBayes in a massive MCMC experiment amounting to 7.5 billions generations. For five different datasets. The above picture describes mean square errors for the probabilities of split, over ten replicates [when meaningful], the worst case being naïve Monte Carlo, with nested sampling and harmonic mean solutions close by. Similar assessments proceed from a comparison of Kullback-Leibler divergences. With the (predicatble?) note that “the methods do a better job approximating the marginal likelihood of more probable trees than less probable trees”. And massive variability for the poorest methods:

The comparison above does not account for time and since some methods are deterministic (and fast) there is little to do about this. The stepping steps solutions are very costly, while on the middle range bridge sampling outdoes path sampling. The assessment of nested sampling found in the conclusion is that it “would appear to be an unwise choice for estimating the marginal likelihoods of topologies, as it produces poor approximate posteriors” (p.12). Concluding at the Gamma Laplus approximation being the winner across all categories! (There is no ABC solution studied in this paper as the model likelihood can be computed in this setup, contrary to our own setting.)

approximative Laplace

Posted in Books, R, Statistics with tags , , , , on August 18, 2018 by xi'an

I came across this question on X validated that wondered about one of our examples in Monte Carlo Statistical Methods. We have included a section on Laplace approximations in the Monte Carlo integration chapter, with a bit of reluctance on my side as this type of integral approximation does not directly connect to Monte Carlo methods. Even less in the case of the example as we aimed at replacing a coverage probability for a Gamma distribution with a formal Laplace approximation. Formal due to the lack of asymptotics, besides the length of the interval (a,b) which probability is approximated. Hence, on top of the typos, the point of the example is not crystal clear, in that it does not show much more than the step-function approximation to the function converges as the interval length gets to zero. For instance, using instead a flat approximation produces an almost as good approximation:

>  xact(5,2,7,9)
[1] 0.1933414
> laplace(5,2,7,9)
[1] 0.1933507
> flat(5,2,7,9)
[1] 0.1953668


What may be more surprising is the resilience of the approximation as the width of the interval increases:

> xact(5,2,5,11)
[1] 0.53366
> lapl(5,2,5,11)
[1] 0.5354954
> plain(5,2,5,11)
[1] 0.5861004