Archive for Bayesian model selection

O’Bayes 2019 conference program

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on May 13, 2019 by xi'an

The full and definitive program of the O’Bayes 2019 conference in Warwick is now on line. Including discussants for all papers. And the three [and free] tutorials on Friday afternoon, 28 June, on model selection (M. Barbieri), MCMC recent advances (G.O. Roberts) and BART (E.I. George). Registration remains open at the reduced rate and submissions of posters can still be sent to me for all conference participants.

Bertrand-Borel debate

Posted in Books, Statistics with tags , , , , , , , , , , , , , on May 6, 2019 by xi'an

On her blog, Deborah Mayo briefly mentioned the Bertrand-Borel debate on the (in)feasibility of hypothesis testing, as reported [and translated] by Erich Lehmann. A first interesting feature is that both [starting with] B mathematicians discuss the probability of causes in the Bayesian spirit of Laplace. With Bertrand considering that the prior probabilities of the different causes are impossible to set and then moving all the way to dismiss the use of probability theory in this setting, nipping the p-values in the bud..! And Borel being rather vague about the solution probability theory has to provide. As stressed by Lehmann.

“The Pleiades appear closer to each other than one would naturally expect. This statement deserves thinking about; but when one wants to translate the phenomenon into numbers, the necessary ingredients are lacking. In order to make the vague idea of closeness more precise, should we look for the smallest circle that contains the group? the largest of the angular distances? the sum of squares of all the distances? the area of the spherical polygon of which some of the stars are the vertices and which contains the others in its interior? Each of these quantities is smaller for the group of the Pleiades than seems plausible. Which of them should provide the measure of implausibility? If three of the stars form an equilateral triangle, do we have to add this circumstance, which is certainly very unlikely apriori, to those that point to a cause?” Joseph Bertrand (p.166)

 

“But whatever objection one can raise from a logical point of view cannot prevent the preceding question from arising in many situations: the theory of probability cannot refuse to examine it and to give an answer; the precision of the response will naturally be limited by the lack of precision in the question; but to refuse to answer under the pretext that the answer cannot be absolutely precise, is to place oneself on purely abstract grounds and to misunderstand the essential nature of the application of mathematics.” Emile Borel (Chapter 4)

Another highly interesting objection of Bertrand is somewhat linked with his conditioning paradox, namely that the density of the observed unlikely event depends on the choice of the statistic that is used to calibrate the unlikeliness, which makes complete sense in that the information contained in each of these statistics and the resulting probability or likelihood differ to an arbitrary extend, that there are few cases (monotone likelihood ratio) where the choice can be made, and that Bayes factors share the same drawback if they do not condition upon the entire sample. In which case there is no selection of “circonstances remarquables”. Or of uniformly most powerful tests.

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.

importance tempering and variable selection

Posted in Books, Statistics with tags , , , , , , , , on November 6, 2018 by xi'an

As reading and commenting the importance tempering for variable selection paper by Giacomo Zanella (previously Warwick) and Gareth Roberts (Warwick) has been on my to-do list for quite a while, the fact that Giacomo presented this work at CIRM Bayesian Masterclass last week was the right nudge to write this post.

The starting point for the method is to simulate from a tempered version of a Gibbs sampler, selecting the component [of the parameter vector θ] according to an importance weight that is the inverse of the conditional posterior to the complementary power. That is, the inverse of the importance weight. This approach differs from classical (MCMC) tempering in that it does not target the original distribution. Hence it produces a weighted sample, whose computing time is of the order of the dimension of θ, even though the tempered simulation of a single conditional can reduce the variance of the estimator. The method is generalisable to any collection of one-component proposal/importance distributions, with the assumption that they have fatter tails that the true conditionals. The resulting Markov chain is reversible with respect to another stationary measure made of the original distribution multiplied by the normalisation factor of the importance weights but this ensures that weighted averages converge to the right quantity. Interestingly so because the powered conditionals are not necessarily coherent from a Gibbsic perspective.

The method is applied to Bayesian [spike-and-slab] variable selection of variables, the importance selection of a subset of covariates being restricted to changing one index at a time. I did not understand first how the computation of the normalising constant avoids involving 2-to-the-power-p terms until Giacomo explained to me that the constant was only computed for conditionals. The complexity gets down from O(|γ|²) to O(|γ|p), where |γ| is the number of variables. Another question I had was about the tempering power β, which selection remains a wee bit of an art!

the Hyvärinen score is back

Posted in pictures, Statistics, Travel with tags , , , , , , , , , , , , , on November 21, 2017 by xi'an

Stéphane Shao, Pierre Jacob and co-authors from Harvard have just posted on arXiv a new paper on Bayesian model comparison using the Hyvärinen score

\mathcal{H}(y, p) = 2\Delta_y \log p(y) + ||\nabla_y \log p(y)||^2

which thus uses the Laplacian as a natural and normalisation-free penalisation for the score test. (Score that I first met in Padova, a few weeks before moving from X to IX.) Which brings a decision-theoretic alternative to the Bayes factor and which delivers a coherent answer when using improper priors. Thus a very appealing proposal in my (biased) opinion! The paper is mostly computational in that it proposes SMC and SMC² solutions to handle the estimation of the Hyvärinen score for models with tractable likelihoods and tractable completed likelihoods, respectively. (Reminding me that Pierre worked on SMC² algorithms quite early during his Ph.D. thesis.)

A most interesting remark in the paper is to recall that the Hyvärinen score associated with a generic model on a series must be the prequential (predictive) version

\mathcal{H}_T (M) = \sum_{t=1}^T \mathcal{H}(y_t; p_M(dy_t|y_{1:(t-1)}))

rather than the version on the joint marginal density of the whole series. (Followed by a remark within the remark that the logarithm scoring rule does not make for this distinction. And I had to write down the cascading representation

\log p(y_{1:T})=\sum_{t=1}^T \log p(y_t|y_{1:t-1})

to convince myself that this unnatural decomposition, where the posterior on θ varies on each terms, is true!) For consistency reasons.

This prequential decomposition is however a plus in terms of computation when resorting to sequential Monte Carlo. Since each time step produces an evaluation of the associated marginal. In the case of state space models, another decomposition of the authors, based on measurement densities and partial conditional expectations of the latent states allows for another (SMC²) approximation. The paper also establishes that for non-nested models, the Hyvärinen score as a model selection tool asymptotically selects the closest model to the data generating process. For the divergence induced by the score. Even for state-space models, under some technical assumptions.  From this asymptotic perspective, the paper exhibits an example where the Bayes factor and the Hyvärinen factor disagree, even asymptotically in the number of observations, about which mis-specified model to select. And last but not least the authors propose and assess a discrete alternative relying on finite differences instead of derivatives. Which remains a proper scoring rule.

I am quite excited by this work (call me biased!) and I hope it can induce following works as a viable alternative to Bayes factors, if only for being more robust to the [unspecified] impact of the prior tails. As in the above picture where some realisations of the SMC² output and of the sequential decision process see the wrong model being almost acceptable for quite a long while…

WBIC, practically

Posted in Statistics with tags , , , , , , , , , on October 20, 2017 by xi'an

“Thus far, WBIC has received no more than a cursory mention by Gelman et al. (2013)”

I had missed this 2015  paper by Nial Friel and co-authors on a practical investigation of Watanabe’s WBIC. Where WBIC stands for widely applicable Bayesian information criterion. The thermodynamic integration approach explored by Nial and some co-authors for the approximation of the evidence, thermodynamic integration that produces the log-evidence as an integral between temperatures t=0 and t=1 of a powered evidence, is eminently suited for WBIC, as the widely applicable Bayesian information criterion is associated with the specific temperature t⁰ that makes the power posterior equidistant, Kullback-Leibler-wise, from the prior and posterior distributions. And the expectation of the log-likelihood under this very power posterior equal to the (genuine) evidence. In fact, WBIC is often associated with the sub-optimal temperature 1/log(n), where n is the (effective?) sample size. (By comparison, if my minimalist description is unclear!, thermodynamic integration requires a whole range of temperatures and associated MCMC runs.) In an ideal Gaussian setting, WBIC improves considerably over thermodynamic integration, the larger the sample the better. In more realistic settings, though, including a simple regression and a logistic [Pima Indians!] model comparison, thermodynamic integration may do better for a given computational cost although the paper is unclear about these costs. The paper also runs a comparison with harmonic mean and nested sampling approximations. Since the integral of interest involves a power of the likelihood, I wonder if a safe version of the harmonic mean resolution can be derived from simulations of the genuine posterior. Provided the exact temperature t⁰ is known…

astronomical evidence

Posted in pictures, Statistics, University life with tags , , , , , , , , , , , , on July 24, 2015 by xi'an

As I have a huge arXiv backlog and an even higher non-arXiv backlog, I cannot be certain I will find time to comment on those three recent and quite exciting postings connecting ABC with astro- and cosmo-statistics [thanks to Ewan for pointing out those to me!]: