Bayes Rules! [book review]

Bayes Rules! is a new introductory textbook on Applied Bayesian Model(l)ing, written by Alicia Johnson (Macalester College), Miles Ott (Johnson & Johnson), and Mine Dogucu (University of California Irvine). Textbook sent to me by CRC Press for review. It is available (free) online as a website and has a github site, as well as a bayesrule R package. (Which reminds me that both our own book R packages, bayess and mcsm, have gone obsolete on CRAN! And that I should find time to figure out the issue for an upgrading…)

As far as I can tell [from abroad and from only teaching students with a math background], Bayes Rules! seems to be catering to early (US) undergraduate students with very little exposure to mathematical statistics or probability, as it introduces basic probability notions like pmf, joint distribution, and Bayes’ theorem (as well as Greek letters!) and shies away from integration or algebra (a covariance matrix occurs on page 437 with a lot . For instance, the Normal-Normal conjugacy derivation is considered a “mouthful” (page 113). The exposition is somewhat stretched along the 500⁺ pages as a result, imho, which is presumably a feature shared with most textbooks at this level, and, accordingly, the exercises and quizzes are more about intuition and reproducing the contents of the chapter than technical. In fact, I did not spot there a mention of sufficiency, consistency, posterior concentration (almost made on page 113), improper priors, ergodicity, irreducibility, &tc., while other notions are not precisely defined, like ESS, weakly informative (page 234) or vague priors (page 77), prior information—which makes the negative answer to the quiz “All priors are informative”  (page 90) rather confusing—, R-hat, density plot, scaled likelihood, and more.

As an alternative to “technical derivations” Bayes Rules! centres on intuition and simulation (yay!) via its bayesrule R package. Itself relying on rstan. Learning from example (as R code is always provided), the book proceeds through conjugate priors, MCMC (Metropolis-Hasting) methods, regression models, and hierarchical regression models. Quite impressive given the limited prerequisites set by the authors. (I appreciated the representations of the prior-likelihood-posterior, especially in the sequential case.)

Regarding the “hot tip” (page 108) that the posterior mean always stands between the prior mean and the data mean, this should be made conditional on a conjugate setting and a mean parameterisation. Defining MCMC as a method that produces a sequence of realisations that are not from the target makes a point, except of course that there are settings where the realisations are from the target, for instance after a renewal event. Tuning MCMC should remain a partial mystery to readers after reading Chapter 7 as the Goldilocks principle is quite vague. Similarly, the derivation of the hyperparameters in a novel setting (not covered by the book) should prove a challenge, even though the readers are encouraged to “go forth and do some Bayes things” (page 509).

While Bayes factors are supported for some hypothesis testing (with no point null), model comparison follows more exploratory methods like X validation and expected log-predictive comparison.

The examples and exercises are diverse (if mostly US centric), modern (including cultural references that completely escape me), and often reflect on the authors’ societal concerns. In particular, their concern about a fair use of the inferred models is preminent, even though a quantitative assessment of the degree of fairness would require a much more advanced perspective than the book allows… (In that respect, Exercise 18.2 and the following ones are about book banning (in the US). Given the progressive tone of the book, and the recent ban of math textbooks in the US, I wonder if some conservative boards would consider banning it!) Concerning the Himalaya submitting running example (Chapters 18 & 19), where the probability to summit is conditional on the age of the climber and the use of additional oxygen, I am somewhat surprised that the altitude of the targeted peak is not included as a covariate. For instance, Ama Dablam (6848 m) is compared with Annapurna I (8091 m), which has the highest fatality-to-summit ratio (38%) of all. This should matter more than age: the Aosta guide Abele Blanc climbed Annapurna without oxygen at age 57! More to the point, the (practical) detailed examples do not bring unexpected conclusions, as for instance the fact that runners [thrice alas!] tend to slow down with age.

A geographical comment: Uluru (page 267) is not a city!, but an impressive sandstone monolith in the heart of Australia, a 5 hours drive away from Alice Springs. And historical mentions: Alan Turing (page 10) and the team at Bletchley Park indeed used Bayes factors (and sequential analysis) in cracking the Enigma, but this remained classified information for quite a while. Arianna Rosenbluth (page 10, but missing on page 165) was indeed a major contributor to Metropolis et al.  (1953, not cited), but would not qualify as a Bayesian statistician as the goal of their algorithm was a characterisation of the Boltzman (or Gibbs) distribution, not statistical inference. And David Blackwell’s (page 10) Basic Statistics is possibly the earliest instance of an introductory Bayesian and decision-theory textbook, but it never mentions Bayes or Bayesianism.

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Book Review section in CHANCE.]

null recurrent = zero utility?

The stability result that the ratio

converges holds for a Harris π-null-recurrent Markov chain for all functions f,g in L¹(π) [Meyn & Tweedie, 1993, Theorem 17.3.2] is rather fascinating. However, it is unclear it can be useful in simulation environments, as for the integral priors we have been studying over the years with Juan Antonio Cano and Diego Salmeron Martinez. Above, the result of an experiment where I simulated a Markov chain as a Normal random walk in dimension one, hence a Harris π-null-recurrent Markov chain for the Lebesgue measure λ, and monitored the stabilisation of the ratio (1) when using two densities for f and g,  to its expected value (1, shown by a red horizontal line). There is quite a variability in the outcome (repeated 100 times),  but the most intriguing is the quick stabilisation of most cumulated averages to values different from 1. Even longer runs display this feature

which I would blame on the excursions of the random walk far away from the central regions for both f and g, that is on long sequences where zeroes keep being added to numerator and denominators in (1). As far as integral approximation is concerned, this is not very helpful!

O’Bayes 19/2

One talk on Day 2 of O’Bayes 2019 was by Ryan Martin on data dependent priors (or “priors”). Which I have already discussed in this blog. Including the notion of a Gibbs posterior about quantities that “are not always defined through a model” [which is debatable if one sees it like part of a semi-parametric model]. Gibbs posterior that is built through a pseudo-likelihood constructed from the empirical risk, which reminds me of Bissiri, Holmes and Walker. Although requiring a prior on this quantity that is  not part of a model. And is not necessarily a true posterior and not necessarily with the same concentration rate as a true posterior. Constructing a data-dependent distribution on the parameter does not necessarily mean an interesting inference and to keep up with the theme of the conference has no automated claim to [more] “objectivity”.

And after calling a prior both Beauty and The Beast!, Erlis Ruli argued about a “bias-reduction” prior where the prior is solution to a differential equation related with some cumulants, connected with an earlier work of David Firth (Warwick).  An interesting conundrum is how to create an MCMC algorithm when the prior is that intractable, with a possible help from PDMP techniques like the Zig-Zag sampler.

While Peter Orbanz’ talk was centred on a central limit theorem under group invariance, further penalised by being the last of the (sun) day, Peter did a magnificent job of presenting the result and motivating each term. It reminded me of the work Jim Bondar was doing in Ottawa in the 1980’s on Haar measures for Bayesian inference. Including the notion of amenability [a term due to von Neumann] I had not met since then. (Neither have I met Jim since the last summer I spent in Carleton.) The CLT and associated LLN are remarkable in that the average is not over observations but over shifts of the same observation under elements of a sub-group of transformations. I wondered as well at the potential connection with the Read Paper of Kong et al. in 2003 on the use of group averaging for Monte Carlo integration [connection apart from the fact that both discussants, Michael Evans and myself, are present at this conference].

ABC with Gibbs steps

With Grégoire Clarté, Robin Ryder and Julien Stoehr, all from Paris-Dauphine, we have just arXived a paper on the specifics of ABC-Gibbs, which is a version of ABC where the generic ABC accept-reject step is replaced by a sequence of n conditional ABC accept-reject steps, each aiming at an ABC version of a conditional distribution extracted from the joint and intractable target. Hence an ABC version of the standard Gibbs sampler. What makes it so special is that each conditional can (and should) be conditioning on a different statistic in order to decrease the dimension of this statistic, ideally down to the dimension of the corresponding component of the parameter. This successfully bypasses the curse of dimensionality but immediately meets with two difficulties. The first one is that the resulting sequence of conditionals is not coherent, since it is not a Gibbs sampler on the ABC target. The conditionals are thus incompatible and therefore convergence of the associated Markov chain becomes an issue. We produce sufficient conditions for the Gibbs sampler to converge to a stationary distribution using incompatible conditionals. The second problem is then that, provided it exists, the limiting and also intractable distribution does not enjoy a Bayesian interpretation, hence may fail to be justified from an inferential viewpoint. We however succeed in producing a version of ABC-Gibbs in a hierarchical model where the limiting distribution can be explicited and even better can be weighted towards recovering the original target. (At least with limiting zero tolerance.)

Hamiltonian MC on discrete spaces

Following a lively discussion with Akihiko Nishimura during a BNP11 poster session last Tuesday, I took the opportunity of the flight to Montréal to read through the arXived paper (written jointly with David Dunson and Jianfeng Liu). The issue is thus one of handling discrete valued parameters in Hamiltonian Monte Carlo. The basic “trick” in handling this complexity goes by turning the discrete support via the inclusion of an auxiliary continuous variable whose discretisation is the discrete parameter, hence resembling to some extent the slice sampler. This removes the discreteness blockage but creates another difficulty, namely handling a discontinuous target density. (I idly wonder why the trick cannot be iterated to second or higher order so that to achieve the right amount of smoothness. Of course, the maths behind would be less cool!) The extension of the Hamiltonian to this setting by a  convolution is a trick I had not seen since the derivation of the Central Limit Theorem during Neveu’s course at Polytechnique.  What I find most exciting in the resolution is the move from a Gaussian momentum to a Laplace momentum, for the reason that I always wondered at alternatives [without trying anything myself!]. The Laplace version is indeed most appropriate here in that it avoids a computation of all discontinuity points and associated values along a trajectory. Since the moves are done component-wise, the method has a Metropolis-within-Gibbs flavour, which actually happens to be a special case. What is also striking is that the approach is both rejection-free and exact, provided ergodicity occurs, which is the case when the stepsize is random.

In addition to this resolution of the discrete parameter problem, the paper presents the further appeal of (re-)running an analysis of the Jolly-Seber capture-recapture model. Where the discrete parameter is the latent number of live animals [or whatever] in the system at any observed time. (Which we cover in Bayesian essentials with R as a neat entry to both dynamic and latent variable models.) I would have liked to see a comparison with the completion approach of Jérôme Dupuis (1995, Biometrika), since I figure the Metropolis version implemented here differs from Jérôme’s. The second example is built on Bissiri et al. (2016) surrogate likelihood (discussed earlier here) and Chopin and Ridgway (2017) catalogue of solutions for not analysing the Pima Indian dataset. (Replaced by another dataset here.)