Archive for model assessment

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.

Assessing models when all models are false

Posted in Statistics, University life with tags , , , , , on November 11, 2010 by xi'an

When I arrived home from Philadelphia, I got the news that John Geweke was giving a seminar at CREST in the early afternoon and thus biked there to attend his talk. The topic was about comparing asset return models, but what interested most was the notion of comparing models without a reference to a true model, a difficulty I have been juggling with for quite a while (at least since the goodness-of-fit paper with Judith Rousseau we never resubmitted!). And for which I still do not find a satisfactory (Bayesian) solution.

Because there is no true model, Durham and Geweke use the daily (possibly Bayesian) predictive

p(y_t|Y^o_{t-1},X^o_{t-1})

as their basis for model assessment and rely on a log scoring rule

\sum_{s=1}^{t-1} \log p_s(y^o_s|Y^o_{s-1},X^o_{s-1})

to compare models. (The ‘o’ in the superscript denotes the observed values.) As reported in the paper this is a proper (or honest) scoring rule. If n models are under competition, a weighted (model) predictive average

\sum_{i=1}^n \,\omega_{s-1;i} p^i_s(y_s|Y^o_{s-1},X^o_{s-1})

can be considered and the paper examines the impact of picking the optimal weight vector (\omega_{t-1,1},\ldots,\omega_{t-1,n}) against the log scoring rule, i.e.

\arg\max_{\mathbf{\omega}_{t-1}} \sum_{s=1}^{t-1} \sum_{i=1}^n \,\omega_{t-1;i} \log p^i_s(y^o_s|Y^o_{s-1},X^o_{s-1})

The weight vector at time t-1 is therefore optimising the backward sequence of predictions of the observed values till time t-1. The interesting empirical result from this study is that, even from a Bayesian perspective, the weights never degenerate, unless one of the models is correct (which is rarely the case!). Thus, even after very long series of observations, the weights of different models remain away from zero (while the Bayesian posterior probability of a single model goes to one). Even though I am not yet at the point of adopting this solution (in particular because it seems to be using the data twice, once through the posterior/predictive and once through the score), I find the approach quite intriguing and hope I can study it further. Maybe a comparison with a Bayesian non-parametric evaluation would make sense…