After a rather intense period of new simulations and versions, Juong Een (Kate) Lee and I have now resubmitted our paper on (some) importance sampling schemes for evidence approximation in mixture models to Bayesian Analysis. There is no fundamental change in the new version but rather a more detailed description of what those importance schemes mean in practice. The original idea in the paper is to improve upon the Rao-Blackwellisation solution proposed by Berkoff et al. (2002) and later by Marin et al. (2005) to avoid the impact of label switching on Chib’s formula. The Rao-Blackwellisation consists in averaging over all permutations of the labels while the improvement relies on the elimination of useless permutations, namely those that produce a negligible conditional density in Chib’s (candidate’s) formula. While the improvement implies truncated the overall sum and hence induces a potential bias (which was the concern of one referee), the determination of the irrelevant permutations after relabelling next to a single mode does not appear to cause any bias, while reducing the computational overload. Referees also made us aware of many recent proposals that conduct to different evidence approximations, albeit not directly related with our purpose. (One was Rodrigues and Walker, 2014, discussed and commented in a recent post.)
Archive for Andrew Gelman
In the latest Sunday Review of the New York Times, the Norwegian novelist Jo Nesbo has a tribune on revenge against misdeeds and law as institutionalized revenge. Somewhat hidden in the current justifications of the legal system(s). (As an aside, he mentions the example of the Icelandic Alþingi where justice was dispensed once a year, resulting in beheadings, stake burnings, and drowning in the pond depicted above…) This came a few days after another tribune on a similar topic by Charles Blow, following the “botched Oklahoma execution of Clayton Lockett”, entitled “Eye-for-eye incivility” (an understatement if any!), and arguing about the economic inefficiency of the death penalty. Besides the basic moral quandaries of taking someone else’s life, perfectly summarised by Franquin in the following dark strip:
This sequence of tribunes links to one of my pet theories, which is that imprisonment is the most inadequate way of addressing crime and law breaking in (modern?) societies. Setting fully aside the moral notions of revenge and punishment, which aim more at the victim or victim’s relatives than at the perpetrator, and of redemption and remorse, which are at best hypothetical and inspired by religious considerations, I do wonder why economists have not tried to come up with more rational and game-theoretic ways of dealing with law-breakers than locking them up all together and expecting them to behave forever after the end of their term. More globally, I find it quite surprising that no one ever seems to question the very notion of sending people to jail. Indeed, it does bring any clear benefit to society as a whole. One of the usual arguments I receive in those occasions is that imprisonment keeps dangerous people away. But that seems a fairly weak notion: (i) most violent offenders are not dangerous in an absolute berserker sense but only because local circumstances made them violent at a given occurrence in space and time, (ii) those offenders are only put away for a while (in most civilised countries), (iii) they are not getting any less dangerous while in prison, and (iv) it does not apply to the vast majority of people jailed. Furthermore, from a pure offer-versus-demand perspective, this may be counterproductive: e.g., putting some thieves away in jail for a while simply gives an opportunity to other thieves to take advantage of the “thieving market”.
The Freakonomics blog has some entries on the topic—somewhat supportive of my notion that most criminals act in an overall rational way for which incentives and decentives could be considered—, but still fails to address the larger picture… I showed this post to Andrew who pointed me (of course!) to his blog, as several entries therein also consider the issue, like this one on the puzzles of criminal justice. Or prison terms for financial fraud? But I would push the argument further and call for an ultimate abolishment of the carceral system, seeking efficient and generalised alternatives to imprisonment. As detailed in this U.N. report I just came across. As I think a time will come when imprisonment will be seen as irrational as witch-burning is considered today.
Here is the second part of my review of Gelman et al.’ Bayesian Data Analysis (third edition):
“When an iterative simulation algorithm is “tuned” (…) the iterations will not in general converge to the target distribution.” (p.297)
Part III covers advanced computation, obviously including MCMC but also model approximations like variational Bayes and expectation propagation (EP), with even a few words on ABC. The novelties in this part are centred at Stan, the language Andrew is developing around Hamiltonian Monte Carlo techniques, a sort of BUGS of the 10’s! (And of course Hamiltonian Monte Carlo techniques themselves. A few (nit)pickings: the book advises important resampling without replacement (p.266) which makes some sense when using a poor importance function but ruins the fundamentals of importance sampling. Plus, no trace of infinite variance importance sampling? of harmonic means and their dangers? In the Metropolis-Hastings algorithm, the proposal is called the jumping rule and denoted by Jt, which, besides giving the impression of a Jacobian, seems to allow for time-varying proposals and hence time-inhomogeneous Markov chains, which convergence properties are much hairier. (The warning comes much later, as exemplified in the above quote.) Moving from “burn-in” to “warm-up” to describe the beginning of an MCMC simulation. Being somewhat 90’s about convergence diagnoses (as shown by the references in Section 11.7), although the book also proposes new diagnoses and relies much more on effective sample sizes. Particle filters are evacuated in hardly half-a-page. Maybe because Stan does not handle particle filters. A lack of intuition about the Hamiltonian Monte Carlo algorithms, as the book plunges immediately into a two-page pseudo-code description. Still using physics vocabulary that put me (and maybe only me) off. Although I appreciated the advice to check analytical gradients against their numerical counterpart.
“In principle there is no limit to the number of levels of variation that can be handled in this way. Bayesian methods provide ready guidance in handling the estimation of the unknown parameters.” (p.381)
I also enjoyed reading the part about modes that stand at the boundary of the parameter space (Section 13.2), even though I do not think modes are great summaries in Bayesian frameworks and while I do not see how picking the prior to avoid modes at the boundary avoids the data impacting the prior, in fine. The variational Bayes section (13.7) is equally enjoyable, with a proper spelled-out illustration, introducing an unusual feature for Bayesian textbooks. (Except that sampling without replacement is back!) Same comments for the Expectation Propagation (EP) section (13.8) that covers brand new notions. (Will they stand the test of time?!)
“Geometrically, if β-space is thought of as a room, the model implied by classical model selection claims that the true β has certain prior probabilities of being in the room, on the floor, on the walls, in the edge of the room, or in a corner.” (p.368)
Part IV is a series of five chapters about regression(s). This is somewhat of a classic, nonetheless Chapter 14 surprised me with an elaborate election example that dabbles in advanced topics like causality and counterfactuals. I did not spot any reference to the g-prior or to its intuitive justifications and the chapter mentions the lasso as a regularisation technique, but without any proper definition of this “popular non-Bayesian form of regularisation” (p.368). In French: with not a single equation! Additional novelty may lie in the numerical prior information about the correlations. What is rather crucially (cruelly?) missing though is a clearer processing of variable selection in regression models. I know Andrew opposes any notion of a coefficient being exactly equal to zero, as ridiculed through the above quote, but the book does not reject model selection, so why not in this context?! Chapter 15 on hierarchical extensions stresses the link with exchangeability, once again. With another neat election example justifying the progressive complexification of the model and the cranks and toggles of model building. (I am not certain the reparameterisation advice on p.394 is easily ingested by a newcomer.) The chapters on robustness (Chap. 17) and missing data (Chap. 18) sound slightly less convincing to me, esp. the one about robustness as I never got how to make robustness agree with my Bayesian perspective. The book states “we do not have to abandon Bayesian principles to handle outliers” (p.436), but I would object that the Bayesian paradigm compels us to define an alternative model for those outliers and the way they are produced. One can always resort to a drudging exploration of which subsample of the dataset is at odds with the model but this may be unrealistic for large datasets and further tells us nothing about how to handle those datapoints. The missing data chapter is certainly relevant to such a comprehensive textbook and I liked the survey illustration where the missing data was in fact made of missing questions. However, I felt the multiple imputation part was not well-presented, fearing readers would not understand how to handle it…
“You can use MCMC, normal approximation, variational Bayes, expectation propagation, Stan, or any other method. But your fit must be Bayesian.” (p.517)
Part V concentrates the most advanced material, with Chapter 19 being mostly an illustration of a few complex models, slightly superfluous in my opinion, Chapter 20 a very short introduction to functional bases, including a basis selection section (20.2) that implements the “zero coefficient” variable selection principle refuted in the regression chapter(s), and does not go beyond splines (what about wavelets?), Chapter 21 a (quick) coverage of Gaussian processes with the motivating birth-date example (and two mixture datasets I used eons ago…), Chapter 22 a more (too much?) detailed study of finite mixture models, with no coverage of reversible-jump MCMC, and Chapter 23 an entry on Bayesian non-parametrics through Dirichlet processes.
“In practice, for well separated components, it is common to remain stuck in one labelling across all the samples that are collected. One could argue that the Gibbs sampler has failed in such a case.” (p.535)
To get back to mixtures, I liked the quote about the label switching issue above, as I was “one” who argued that the Gibbs sampler fails to converge! The corresponding section seems to favour providing a density estimate for mixture models, rather than component-wise evaluations, but it nonetheless mentions the relabelling by permutation approach (if missing our 2000 JASA paper). The section about inferring on the unknown number of components suggests conducting a regular Gibbs sampler on a model with an upper bound on the number of components and then checking for empty components, an idea I (briefly) considered in the mid-1990’s before the occurrence of RJMCMC. Of course, the prior on the components matters and the book suggests using a Dirichlet with fixed sum like 1 on the coefficients for all numbers of components.
“14. Objectivity and subjectivity: discuss the statement `People tend to believe results that support their preconceptions and disbelieve results that surprise them. Bayesian methods tend to encourage this undisciplined mode of thinking.’¨ (p.100)
Obviously, this being a third edition begets the question, what’s up, doc?!, i.e., what’s new [when compared with the second edition]? Quite a lot, even though I am not enough of a Gelmanian exegist to produce a comparision table. Well, for a starter, David Dunson and Aki Vethtari joined the authorship, mostly contributing to the advanced section on non-parametrics, Gaussian processes, EP algorithms. Then the Hamiltonian Monte Carlo methodology and Stan of course, which is now central to Andrew’s interests. The book does include a short Appendix on running computations in R and in Stan. Further novelties were mentioned above, like the vision of weakly informative priors taking over noninformative priors but I think this edition of Bayesian Data Analysis puts more stress on clever and critical model construction and on the fact that it can be done in a Bayesian manner. Hence the insistence on predictive and cross-validation tools. The book may be deemed somewhat short on exercices, providing between 3 and 20 mostly well-developed problems per chapter, often associated with datasets, rather than the less exciting counter-example above. Even though Andrew disagrees and his students at ENSAE this year certainly did not complain, I personally feel a total of 220 exercices is not enough for instructors and self-study readers. (At least, this reduces the number of email requests for solutions! Esp. when 50 of those are solved on the book website.) But this aspect is a minor quip: overall this is truly the reference book for a graduate course on Bayesian statistics and not only Bayesian data analysis.
Andrew Gelman and his coauthors, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Don Rubin, have now published the latest edition of their book Bayesian Data Analysis. David and Aki are newcomers to the authors’ list, with an extended section on non-linear and non-parametric models. I have been asked by Sam Behseta to write a review of this new edition for JASA (since Sam is now the JASA book review editor). After wondering about my ability to produce an objective review (on the one hand, this is The Competition to Bayesian Essentials!, on the other hand Andrew is a good friend spending the year with me in Paris), I decided to jump for it and write a most subjective review, with the help of Clara Grazian who was Andrew’s teaching assistant this year in Paris and maybe some of my Master students who took Andrew’s course. The second edition was reviewed in the September 2004 issue of JASA and we now stand ten years later with an even more impressive textbook. Which truly what Bayesian data analysis should be.
This edition has five parts, Fundamentals of Bayesian Inference, Fundamentals of Bayesian Data Analysis, Advanced Computation, Regression Models, and Non-linear and Non-parametric Models, plus three appendices. For a total of xiv+662 pages. And a weight of 2.9 pounds (1395g on my kitchen scale!) that makes it hard to carry around in the metro…. I took it to Warwick (and then Nottingham and Oxford and back to Paris) instead.
“We could avoid the mathematical effort of checking the integrability of the posterior density (…) The result would clearly show the posterior contour drifting off toward infinity.” (p.111)
While I cannot go into a detailed reading of those 662 pages (!), I want to highlight a few gems. (I already wrote a detailed and critical analysis of Chapter 6 on model checking in that post.) The very first chapter provides all the necessary items for understanding Bayesian Data Analysis without getting bogged in propaganda or pseudo-philosophy. Then the other chapters of the first part unroll in a smooth way, cruising on the B highway… With the unique feature of introducing weakly informative priors (Sections 2.9 and 5.7), like the half-Cauchy distribution on scale parameters. It may not be completely clear how weak a weakly informative prior, but this novel notion is worth including in a textbook. Maybe a mild reproach at this stage: Chapter 5 on hierarchical models is too verbose for my taste, as it essentially focus on the hierarchical linear model. Of course, this is an essential chapter as it links exchangeability, the “atom” of Bayesian reasoning used by de Finetti, with hierarchical models. Still. Another comment on that chapter: it broaches on the topic of improper posteriors by suggesting to run a Markov chain that can exhibit improperness by enjoying an improper behaviour. When it happens as in the quote above, fine!, but there is no guarantee this is always the case! For instance, improperness may be due to regions near zero rather than infinity. And a last barb: there is a dense table (Table 5.4, p.124) that seems to run contrariwise to Andrew’s avowed dislike of tables. I could also object at the idea of a “true prior distribution” (p.128), or comment on the trivia that hierarchical chapters seem to attract rats (as I also included a rat example in the hierarchical Bayes chapter of Bayesian Choice and so does the BUGS Book! Hence, a conclusion that Bayesian textbooks are better be avoided by muriphobiacs…)
“Bayes factors do not work well for models that are inherently continuous (…) Because we emphasize continuous families of models rather than discrete choices, Bayes factors are rarely relevant in our approach to Bayesian statistics.” (p.183 & p.193)
Part II is about “the creative choices that are required, first to set up a Bayesian model in a complex problem, then to perform the model checking and confidence building that is typically necessary to make posterior inferences scientifically defensible” (p.139). It is certainly one of the strengths of the book that it allows for a critical look at models and tools that are rarely discussed in more theoretical Bayesian books. As detailed in my earlier post on Chapter 6, model checking is strongly advocated, via posterior predictive checks and… posterior predictive p-values, which are at best empirical indicators that something could be wrong, definitely not that everything’s allright! Chapter 7 is the model comparison equivalent of Chapter 6, starting with the predictive density (aka the evidence or the marginal likelihood), but completely bypassing the Bayes factor for information criteria like the Watanabe-Akaike or widely available information criterion (WAIC), and advocating cross-validation, which is empirically satisfying but formally hard to integrate within a full Bayesian perspective. Chapter 8 is about data collection, sample surveys, randomization and related topics, another entry that is missing from most Bayesian textbooks, maybe not that surprising given the research topics of some of the authors. And Chapter 9 is the symmetric in that it focus on the post-modelling step of decision making.
(Second part of the review to appear on Monday, leaving readers the weekend to recover!)
We just submitted a letter to PNAS with Andrew Gelman last week, in reaction to Val Johnson’s recent paper “Revised standards for statistical evidence”, essentially summing up our earlier comments within 500 words. Actually, we wrote one draft each! In particular, Andrew came up with the (neat) rhetorical idea of alternative Ronald Fishers living in parallel universes who had each set a different significance reference level and for whom alternative Val Johnsons would rise and propose a modification of the corresponding Fisher’s level. For which I made the above graph, left out of the letter and its 500 words. It relates “the old z” and “the new z”, meaning the boundaries of the rejection zones when, for each golden dot, the “old z” is the previous “new z” and “the new z” is Johnson’s transform. We even figured out that Val’s transform was bringing the significance down by a factor of 10 in a large range of values. As an aside, we also wondered why most of the supplementary material was spent on deriving UMPBTs for specific (formal) problems when the goal of the paper sounded much more global…
As I am aware we are not the only ones to have submitted a letter about Johnson’s proposal, I am quite curious at the reception we will get from the editor! (Although I have to point out that all of my earlier submissions of letters to to PNAS got accepted.)
Andrew Gelman will be visiting Paris-Dauphine and CREST next academic year, with support from those institutions as well as CNRS and Ville de Paris). Which is why he is learning how to pronounce Le loup est revenu. (Maybe not why, as this is not the most useful sentence in downtown Paris…) Very exciting news for all of us local Bayesians (or bayésiens). In addition, Andrew will teach from the latest edition of his book Bayesian Data Analysis, co-authored by John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Don Rubin. He will actually start teaching mi-October, which means the book will not be out yet: so the students at Paris-Dauphine and ENSAE will get a true avant-première of Bayesian Data Analysis. Of course, this item of information will be sadistically tantalising to ‘Og’s readers who cannot spend the semester in Paris. For those who can, I presume there is a way to register for the course as auditeur libre at either Paris-Dauphine or ENSAE.
You assume that I am interested in long-term average properties of procedures, even though I have so often argued that they are at most necessary (as consequences of good procedures), but scarcely sufficient for a severity assessment. The error statistical account I have developed is a statistical philosophy. It is not one to be found in Neyman and Pearson, jointly or separately, except in occasional glimpses here and there (unfortunately). It is certainly not about well-defined accept-reject rules. If N-P had only been clearer, and Fisher better behaved, we would not have had decades of wrangling. However, I have argued, the error statistical philosophy explicates, and directs the interpretation of, frequentist sampling theory methods in scientific, as opposed to behavioural, contexts. It is not a complete philosophy…but I think Gelmanian Bayesians could find in it a source of “standard setting”.
You say “the prior is both a probabilistic object, standard from this perspective, and a subjective construct, translating qualitative personal assessments into a probability distribution. The extension of this dual nature to the so-called “conventional” priors (a very good semantic finding!) is to set a reference … against which to test the impact of one’s prior choices and the variability of the resulting inference. …they simply set a standard against which to gauge our answers.”
I think there are standards for even an approximate meaning of “standard-setting” in science, and I still do not see how an object whose meaning and rationale may fluctuate wildly, even in a given example, can serve as a standard or reference. For what?
Perhaps the idea is that one can gauge how different priors change the posteriors, because, after all, the likelihood is well-defined. That is why the prior and not the likelihood is the camel. But it isn’t obvious why I should want the camel. (camel/gnat references in the paper and response).