Approximation Methods in Bayesian Analysis [#1]

Interesting first day, if somewhat intense!, with Sylvia Richardson talking of divide & conquer strategies, which we had recently explored in our work on mixtures with Adrien and Judith. Then Jere Koskela (Warwick) on sufficient conditions for consistent inference on trees like Kingman’s coalescent. And in the afternoon Julia-Adela Palacios on tree estimation, with novel notions of tree distances and barycentre (soon to appear in Biometrika!). And Julyan Arbel on the notion and properties of priors within (deep) neural networks, following different asymptotics. On top of this, two fast sessions for people presenting their poster, to be attended tonight. And the usual early morning run to swim 25mn in one of the calanques… Luckily enough, Marseille is not too hot this week, feeling even nicer than Paris!

21w5107 [½day 4]

Final ½ day of the 21w5107 workshop for me, as our initial plans were to stop today due to the small number of participants on site. And I had booked plane tickets early, too early. I will thus sadly miss the four afternoon talks, mea culpa! However I did attend Noiritt Chandra’s talk on Bayesian factor analysis. Which has always been a bit of a mystery to me in the sense that the number q of factors need be specified, which is a prior input one rarely controls. Here the goal is to estimate a covariance matrix with a sparse representation. And q is estimated by empirical likelihood ahead of the estimation of the matrix. The focus was on minimaxity and MCMC implementation rather than objective Bayes per se! Then, Daniele Durante spoke about analytical posteriors for probit models using unified skew-Normal priors (following a 2019 Biometrika paper). Including marginal posteriors and marginal likelihood. And for various extensions like dynamic probit models. Opening other computational issues such as simulating high dimensional truncated Normal distributions. (Potential use of delayed acceptance there?) This second talk was also drifting away from objective Bayes! In the first half of his talk, Filippo Ascolani introduced us to trees of random probability measures, each mother node being the distribution of the atoms of the children nodes. (Interestingly, Kingman is both connected to (coalescent) trees and to completely random measures.) My naïve first impression was that the distributions would get more and more degenerate as the number of levels in the tree would increase, however I am unsure this is correct as Filippo mentioned getting observations on all nodes. The talk also made me wonder at how this could be related Radford Neal’s Dirichlet trees. (Which I discovered at my first ICMS workshop about 20 years ago.) Yang Ni concluded the morning with a talk on causality that provided (to me) a very smooth (re)introduction to Bayesian causal graphs.

Even more than last time, I enormously enjoyed the workshop, its location, the fantastic staff at the hotel, and the reconnection with dear friends!, just regretting we could not be a few more. I appreciate the efforts made by on-line participants to stay connected and intervene (thanks, Ed!), but the quality of interactions is sadly of another magnitude when spending all our time together. Hopefully there will be a next time and hopefully we’ll then be back to larger size (and hopefully the location will remain the same). Hasta luego, Oaxaca!

prime suspects [book review]

I was contacted by Princeton University Press to comment on the comic book/graphic novel Prime Suspects (The Anatomy of Integers and Permutations), by Andrew Granville (mathematician) & Jennifer Granville (writer), and Robert Lewis (illustrator), and they sent me the book. I am not a big fan of graphic book entries to mathematical even less than to statistical notions (Logicomix being sort of an exception for its historical perspective and nice drawing style) and this book did nothing to change my perspective on the subject. First, the plot is mostly a pretense at introducing number theory concepts and I found it hard to follow it for more than a few pages. The [noires maths] story is that “forensic maths” detectives are looking at murders that connects prime integers and permutations… The ensuing NCIS-style investigation gives the authors the opportunity to skim through the whole cenacle of number theorists, plus a few other mathematicians, who appear as more or less central characters. Even illusory ones like Nicolas Bourbaki. And Alexander Grothendieck as a recluse and clairvoyant hermit [who in real life did not live in a Pyrénées cavern!!!]. Second, I [and nor is Andrew who was in my office when the book arrived!] am not particularly enjoying the drawings or the page composition or the colours of this graphic novel, especially because I find the characters drawn quite inconsistently from one strip to the next, to the point of being unrecognisable, and, if it matters, hardly resembling their real-world equivalent (as seen in the portrait of Persi Diaconis). To be completely honest, the drawings look both ugly and very conventional to me, in that I do not find much of a characteristic style to them. To contemplate what Jacques Tardi,  François Schuiten or José Muñoz could have achieved with the same material… (Or even Edmond Baudoin, who drew the strips for the graphic novels he coauthored with Cédric Villani.) The graphic novel (with a prime 181 pages) is postfaced with explanations about the true persons behind the characters, from Carl Friedriech Gauß to Terry Tao, and of course on the mathematical theory for the analogies between the prime and cycles frequencies behind the story. Which I find much more interesting and readable, obviously. (With a surprise appearance of Kingman’s coalescent!) But also somewhat self-defeating in that so much has to be explained on the side for the links between the story, the characters and the background heavily loaded with “obscure references” to make sense to more than a few mathematician readers. Who may prove to be the core readership of this book.

There is also a bit of a Gödel-Escher-and-Bach flavour in that a piece by Robert Schneider called Réverie in Prime Time Signature is included, while an Escher’s infinite stairway appears in one page, not far from what looks like Milano Vittorio Emmanuelle gallery (On the side, I am puzzled by the footnote on p.208 that “I should clarify that selecting a random permutation and a random prime, as described, can be done easily, quickly, and correctly”. This may be connected to the fact that the description of Bach’s algorithm provided therein is incomplete.)

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]

JSM 2015 [day #3]

My first morning session was about inference for philogenies. While I was expecting some developments around Kingman’s  coalescent models my coauthors needed and developped ABC for, I was surprised to see models that were producing closed form (or close enough to) likelihoods. Due to strong restrictions on the population sizes and migration possibilities, as explained later to me by Vladimir Minin. No need for ABC there since MCMC was working on the species trees, with Vladimir Minin making use of [the Savage Award winner] Vinayak Rao’s approach on trees that differ from the coalescent. And enough structure to even consider and demonstrate tree identifiability in Laura Kubatko’s case.

I then stopped by the astrostatistics session as the first talk by Gwendolin Eddie was about galaxy mass estimation, a problem I may actually be working on in the Fall, but it ended up being a completely different problem and I was further surprised that the issue of whether or not the data was missing at random was not considered by the authors.

Christening a session Unifying foundation(s) may be calling for trouble, at least from me! In this spirit, Xiao Li Meng gave a talk attempting at a sort of unification of the frequentist, Bayesian, and fiducial paradigms by introducing the notion of personalized inference, which is a notion I had vaguely thought of in the past. How much or how far do you condition upon? However, I have never thought of this justifying fiducial inference in any way and Xiao Li’s lively arguments during and after the session not enough to convince me of the opposite: Prior-free does not translate into (arbitrary) choice-free. In the earlier talk about confidence distributions by Regina Liu and Minge Xie, that I partly missed for Galactic reasons, I just entered into the room at the very time when ABC was briefly described as a confidence distribution because it was not producing a convergent approximation to the exact posterior, a logic that escapes me (unless those confidence distributions are described in such a loose way as to include about any method f inference). Dongchu Sun also gave us a crash course on reference priors, with a notion of random posteriors I had not heard of before… As well as constructive posteriors… (They seemed to mean constructible matching priors as far as I understood.)

The final talk in this session by Chuanhei Liu on a new approach (modestly!) called inferential model was incomprehensible, with the speaker repeatedly stating that the principles were too hard to explain in five minutes and needed an incoming book… I later took a brief look at an associated paper, which relates to fiducial inference and to Dempster’s belief functions. For me, it has the same Münchhausen feeling of creating a probability out of nothing, creating a distribution on the parameter by ignoring the fact that the fiducial equation x=a(θ,u) modifies the distribution of u once x is observed.

posterior predictive checks for admixture models

In a posting coincidence, just a few days after we arXived our paper on ABC model choice with random forests, where we use posterior predictive errors for assessing the variability of the random forest procedure, David Mimno, David Blei, and Barbara Engelhardt arXived a paper on posterior predictive checks to quantify lack-of-fit in admixture models of latent population structure, which deals with similar data and models, while also using the posterior predictive as a central tool. (Marginalia: the paper is a wee bit difficult to read [esp. with France-Germany playing in the airport bar!] as the modelling is only clearly described at the very end. I suspect this arXived version was put together out of a submission to a journal like Nature or PNAS, with mentions of a Methods section that does not appear here and of Supplementary Material that turned into subsections of the Discussion section.)

The dataset are genomic datasets made of SNPs (single nucleotide polymorphisms). For instance, the first (HapMap) dataset corresponds to 1,043 individuals and 468,167 SNPs. The model is simpler than Kingman’s coalescent, hence its likelihood does not require ABC steps to run inference. The admixture model in the paper is essentially a mixture model over ancestry indices with individual dependent weights with Bernoulli observations, hence resulting into a completed likelihood of the form

(which looks more formidable than it truly is!). Regular Bayesian inference is thus possible in this setting, implementing e.g. Gibbs sampling. The authors chose instead to rely on EM and thus derived the maximum likelihood estimators of the (many) parameters of the admixture. And of the latent variables z. Their posterior predictive check is based on the simulation of pseudo-observations (as in ABC!) from the above likelihood, with parameters and latent variables replaced with their EM estimates (unlike ABC). There is obviously some computational reason in doing this instead of simulating from the posterior, albeit implicit in the paper. I am however slightly puzzled by the conditioning on the latent variable estimate ẑ, as its simulation is straightforward and as a latent variable is more a missing observation than a parameter. Given those 30 to 100 replications of the data, an empirical distribution of a discrepancy function is used to assess whether or not the equivalent discrepancy for the observation is an outlier. If so, the model is not appropriate for the data. (Interestingly, the discrepancy is measured via the Bayes factor of z-scores.)

The connection with our own work is that the construction of discrepancy measures proposed in this paper could be added to our already large collection of summary statistics to check to potential impact in model comparison, i.e. for a significant contribution to the random forest nodes.  Conversely, the most significant summary statistics could then be tested as discrepancy measures. Or, more in tune with our Series B paper on the proper selection of summary variables, the distribution of those discrepancy measures could be compared across potential models. Assuming this does not take too much computing power…