Sancerre

prior sensitivity of the marginal likelihood

Fernando Llorente and (Madrilene) coauthors have just arXived a paper on the safe use of prior densities for Bayesian model selection. Rather than blaming the Bayes factor, or excommunicating some improper priors, they consider in this survey solutions to design “objective” priors in model selection. (Writing this post made me realised I had forgotten to arXive a recent piece I wrote on the topic, based on short courses and blog pieces, for an incoming handbook on Bayesian advance(ment)s! Soon to be corrected.)

While intrinsically interested in the topic and hence with the study, I somewhat disagree with the perspective adopted by the authors. They for instance stick to the notion that a flat prior over the parameter space is appropriate as “the maximal expression of a non-informative prior” (despite depending on the parameterisation). Over bounded sets at least, while advocating priors “with great scale parameter” otherwise. They also refer to Jeffreys (1939) priors, by which they mean estimation priors rather than testing priors. As uncovered by Susie Bayarri and Gonzalo Garcia-Donato. Considering asymptotic consistency, they state that “in the asymptotic regime, Bayesian model selection is more sensitive to the sample size D than to the prior specifications”, which I find both imprecise and confusing,  as my feeling is that the prior specification remains overly influential as the sample size increases. (In my view, consistency is a minimalist requirement, rather than “comforting”.) The argument therein that a flat prior is informative for model choice stems from the fact that the marginal likelihood goes to zero as the support of the prior goes to infinity, which may have been an earlier argument of Jeffreys’ (1939), but does not carry much weight as the property is shared by many other priors (as remarked later). Somehow, the penalisation aspect of the marginal is not exploited more deeply in the paper. In the “objective” Bayes section, they adhere to the (convenient but weakly supported) choice of a common prior on the nuisance parameters (shared by different models). Their main argument is to develop (heretic!) “data-based priors”, from Aitkin (1991, not cited) double use of the data (or setting the likelihood to the power two), all the way to the intrinsic and fractional Bayes factors of Tony O’Hagan (1995), Jim Berger and Luis Pericchi (1996), and to the expected posterior priors of Pérez and Berger (2002) on which I worked with Juan Cano and Diego Salmeròn. (While the presentation is made against a flat prior, nothing prevents the use of another reference, improper, prior.) A short section also mentions the X-validation approach(es) of Aki Vehtari and co-authors.

Jeffreys priors for hypothesis testing [Bayesian reads #2]

A second (re)visit to a reference paper I gave to my OxWaSP students for the last round of this CDT joint program. Indeed, this may be my first complete read of Susie Bayarri and Gonzalo Garcia-Donato 2008 Series B paper, inspired by Jeffreys’, Zellner’s and Siow’s proposals in the Normal case. (Disclaimer: I was not the JRSS B editor for this paper.) Which I saw as a talk at the O’Bayes 2009 meeting in Phillie.

The paper aims at constructing formal rules for objective proper priors in testing embedded hypotheses, in the spirit of Jeffreys’ Theory of Probability “hidden gem” (Chapter 3). The proposal is based on symmetrised versions of the Kullback-Leibler divergence κ between null and alternative used in a transform like an inverse power of 1+κ. With a power large enough to make the prior proper. Eventually multiplied by a reference measure (i.e., the arbitrary choice of a dominating measure.) Can be generalised to any intrinsic loss (not to be confused with an intrinsic prior à la Berger and Pericchi!). Approximately Cauchy or Student’s t by a Taylor expansion. To be compared with Jeffreys’ original prior equal to the derivative of the atan transform of the root divergence (!). A delicate calibration by an effective sample size, lacking a general definition.

At the start the authors rightly insist on having the nuisance parameter v to differ for each model but… as we all often do they relapse back to having the “same ν” in both models for integrability reasons. Nuisance parameters make the definition of the divergence prior somewhat harder. Or somewhat arbitrary. Indeed, as in reference prior settings, the authors work first conditional on the nuisance then use a prior on ν that may be improper by the “same” argument. (Although conditioning is not the proper term if the marginal prior on ν is improper.)

The paper also contains an interesting case of the translated Exponential, where the prior is L¹ Student’s t with 2 degrees of freedom. And another one of mixture models albeit in the simple case of a location parameter on one component only.

ABC variable selection

Prior to the ISBA 2018 meeting, Yi Liu, Veronika Ročková, and Yuexi Wang arXived a paper on relying ABC for finding relevant variables, which is a very original approach in that ABC is not as much the object as it is a tool. And which Veronika considered during her Susie Bayarri lecture at ISBA 2018. In other words, it is not about selecting summary variables for running ABC but quite the opposite, selecting variables in a non-linear model through an ABC step. I was going to separate the two selections into algorithmic and statistical selections, but it is more like projections in the observation and covariate spaces. With ABC still providing an appealing approach to approximate the marginal likelihood. Now, one may wonder at the relevance of ABC for variable selection, aka model choice, given our warning call of a few years ago. But the current paper does not require low-dimension summary statistics, hence avoids the difficulty with the “other” Bayes factor.

In the paper, the authors consider a spike-and… forest prior!, where the Bayesian CART selection of active covariates proceeds through a regression tree, selected covariates appearing in the tree and others not appearing. With a sparsity prior on the tree partitions and this new ABC approach to select the subset of active covariates. A specific feature is in splitting the data, one part to learn about the regression function, simulating from this function and comparing with the remainder of the data. The paper further establishes that ABC Bayesian Forests are consistent for variable selection.

“…we observe a curious empirical connection between π(θ|x,ε), obtained with ABC Bayesian Forests  and rescaled variable importances obtained with Random Forests.”

The difference with our ABC-RF model choice paper is that we select summary statistics [for classification] rather than covariates. For instance, in the current paper, simulation of pseudo-data will depend on the selected subset of covariates, meaning simulating a model index, and then generating the pseudo-data, acceptance being a function of the L² distance between data and pseudo-data. And then relying on all ABC simulations to find which variables are in more often than not to derive the median probability model of Barbieri and Berger (2004). Which does not work very well if implemented naïvely. Because of the immense size of the model space, it is quite hard to find pseudo-data close to actual data, resulting in either very high tolerance or very low acceptance. The authors get over this difficulty by a neat device that reminds me of fractional or intrinsic (pseudo-)Bayes factors in that the dataset is split into two parts, one that learns about the posterior given the model index and another one that simulates from this posterior to compare with the left-over data. Bringing simulations closer to the data. I do not remember seeing this trick before in ABC settings, but it is very neat, assuming the small data posterior can be simulated (which may be a fundamental reason for the trick to remain unused!). Note that the split varies at each iteration, which means there is no impact of ordering the observations.

Altos de Losada [guest wine post by Susie]

[Here is a wine criticism written by Susie Bayarri in 2013 about a 2008 bottle of Altos de Losada, a wine from Leon:]

The cork is fantastic. Very good presentation and labelling of the bottle. The wine  color is like dark cherry, I would almost say of the color of blood. Very bright although unfiltered. The cover is d16efinitely high. The tear is very nice (at least in my glass), slow, wide, through parallel streams… but it does not dye my glass at all.

The bouquet is its best feature… it is simply voluptuous… with ripe plums as well as vanilla, some mineral tone plus a smoky hint. I cannot quite detect which wood is used… I have always loved the bouquet of this wine…

In mouth, it remains a bit closed. Next time, I will make sure I decant it (or I will use that Venturi device) but it is nonetheless excellent… the wine is truly fruity, but complex as well (nothing like grape juice). The tannins are definitely present, but tamed and assimilated (I think they will continue to mellow) and it has just a hint of acidity… Despite its alcohol content, it remains light, neither overly sweet nor heavy. The after-taste offers a pleasant bitterness… It is just delicious, an awesome wine!