21w5107 [day 2]

After a rich and local (if freezing) dinner on a rooftop facing the baroque Oaxaca cathedral, and an early invigorating outdoor swim in my case!, the morning session was mostly on mixtures, with Helen Ogden exploring X validation for (estimating the number k of components for) finite mixtures, when using the likelihood as an objective function. I was unclear of the goal however when considering that the data supporting the study was Uniform (0,1), nothing like a mixture of Normal distributions. And about the consistency attached to the objective function. The session ended with Diana Cai presenting a counter-argument in the sense that she proved, along with Trevor Campbell and Tamara Broderick, that the posterior on k diverges to infinity with the number n of observations if a mixture model is misspecified for said data. Which does not come as a major surprise since there is no properly defined value of k when the data is not generated from the adopted mixture. I would love to see an extension to the case when the k component mixture contains a non-parametric component! In-between, Alexander Ly discussed Bayes factors for multiple datasets, with some asymptotics showing consistency for some (improper!) priors if one sample size grows to infinity. With actually attaining the same rate under both hypotheses. Luis Nieto-Barajas presented an approach on uncertainty assessment through KL divergence for random probability measures, which requires a calibration of the KL in this setting, as KL does not enjoy a uniform scale, and a prior on a Pólya tree. And Chris Holmes presented a recent work with Edwin Fong and Steven Walker on a prediction approach to Bayesian inference. Which I had had on my reading list for a while. It is a very original proposal where likelihoods and priors are replaced by the sequence of posterior predictives and only parameters of interest get simulated. The Bayesian flavour of the approach is delicate to assess though, albeit a form of non-parametric Bayesian perspective… (I still need to read the paper carefully.)

In the afternoon session, Judith Rousseau presented her recent foray in cut posteriors for semi-parametric HMMs. With interesting outcomes for efficiently estimating the transition matrix, the component distributions, and the smoothing distribution. I wonder at the connection with safe Bayes in that cut posteriors induce a loss of information. Sinead Williamson spoke on distributed MCMC for BNP. Going back at the “theme of the day”, namely clustering and finding the correct (?) number of clusters. With a collapsed versus uncollapsed division that reminded me of the marginal vs. conditional María Gil-Leyva discussed yesterday. Plus a decomposition of a random measure into a finite mixture and an infinite one that also reminded me of the morning talk of Diana Cai. (And making me wonder at the choice of the number K of terms in the finite part.) Michele Guindani spoke about clustering distributions (with firecrackers as a background!). Using the nDP mixture model, which was show to suffer from degeneracy (as discussed by Frederico Camerlenghi et al. in BA). The subtle difference stands in using the same (common) atoms in all random distributions at the top of the hierarchy, with independent weights. Making the partitions partially exchangeable. The approach relies on Sylvia’s generalised mixtures of finite mixtures. With interesting applications to microbiome and calcium imaging (including a mice brain in action!). And Giovanni Rebaudo presented a generalised notion of clustering aligned on a graph, with some observations located between the nodes corresponding to clusters. Represented as a random measure with common parameters for the clusters and separated parameters outside. Interestingly playing on random partitions, Pólya urns, and species sampling.

slice sampling for Dirichlet mixture process

When working with my PhD student Changye in Dauphine this morning I realised that slice sampling also applies to discrete support distributions and could even be of use in such settings. That it works is (now) straightforward in that the missing variable representation behind the slice sampler also applies to densities defined with respect to a discrete measure. That this is useful transpires from the short paper of Stephen Walker (2007) where we saw this, as Stephen relies on the slice sampler to sample from the Dirichlet mixture model by eliminating the tail problem associated with this distribution. (This paper appeared in Communications in Statistics and it is through Pati & Dunson (2014) taking advantage of this trick that Changye found about its very existence. I may have known about it in an earlier life, but I had clearly forgotten everything!)

While the prior distribution (of the weights) of the Dirichlet mixture process is easy to generate via the stick breaking representation, the posterior distribution is trickier as the weights are multiplied by the values of the sampling distribution (likelihood) at the corresponding parameter values and they cannot be normalised. Introducing a uniform to replace all weights in the mixture with an indicator that the uniform is less than those weights corresponds to a (latent variable) completion [or a demarginalisation as we called this trick in Monte Carlo Statistical Methods]. As elaborated in the paper, the Gibbs steps corresponding to this completion are easy to implement, involving only a finite number of components. Meaning the allocation to a component of the mixture can be operated rather efficiently. Or not when considering that the weights in the Dirichlet mixture are not monotone, hence that a large number of them may need to be computed before picking the next index in the mixture when the uniform draw happens to be quite small.

comments on Watson and Holmes

“The world is full of obvious things which nobody by any chance ever observes.” The Hound of the Baskervilles

In connection with the incoming publication of James Watson’s and Chris Holmes’ Approximating models and robust decisions in Statistical Science, Judith Rousseau and I wrote a discussion on the paper that has been arXived yesterday.

“Overall, we consider that the calibration of the Kullback-Leibler divergence remains an open problem.” (p.18)

While the paper connects with earlier ones by Chris and coauthors, and possibly despite the overall critical tone of the comments!, I really appreciate the renewed interest in robustness advocated in this paper. I was going to write Bayesian robustness but to differ from the perspective adopted in the 90’s where robustness was mostly about the prior, I would say this is rather a Bayesian approach to model robustness from a decisional perspective. With definitive innovations like considering the impact of posterior uncertainty over the decision space, uncertainty being defined e.g. in terms of Kullback-Leibler neighbourhoods. Or with a Dirichlet process distribution on the posterior. This may step out of the standard Bayesian approach but it remains of definite interest! (And note that this discussion of ours [reluctantly!] refrained from capitalising on the names of the authors to build easy puns linked with the most Bayesian of all detectives!)

Dirichlet process mixture inconsistency

Judith Rousseau pointed out to me this NIPS paper by Jeff Miller and Matthew Harrison on the possible inconsistency of Dirichlet mixtures priors for estimating the (true) number of components in a (true) mixture model. The resulting posterior on the number of components does not concentrate on the right number of components. Which is not the case when setting a prior on the unknown number of components of a mixture, where consistency occurs. (The inconsistency results established in the paper are actually focussed on iid Gaussian observations, for which the estimated number of Gaussian components is almost never equal to 1.) In a more recent arXiv paper, they also show that a Dirichlet prior on the weights and a prior on the number of components can still produce the same features as a Dirichlet mixtures priors. Even the stick breaking representation! (Paper that I already reviewed last Spring.)

Conditional love [guest post]

[When Dan Simpson told me he was reading Terenin’s and Draper’s latest arXival in a nice Bath pub—and not a nice bath tub!—, I asked him for a blog entry and he agreed. Here is his piece, read at your own risk! If you remember to skip the part about Céline Dion, you should enjoy it very much!!!]

Probability has traditionally been described, as per Kolmogorov and his ardent follower Katy Perry, unconditionally. This is, of course, excellent for those of us who really like measure theory, as the maths is identical. Unfortunately mathematical convenience is not necessarily enough and a large part of the applied statistical community is working with Bayesian methods. These are unavoidably conditional and, as such, it is natural to ask if there is a fundamentally conditional basis for probability.

Bruno de Finetti—and later Richard Cox and Edwin Jaynes—considered conditional bases for Bayesian probability that are, unfortunately, incomplete. The critical problem is that they mainly consider finite state spaces and construct finitely additive systems of conditional probability. For a variety of reasons, neither of these restrictions hold much truck in the modern world of statistics.

In a recently arXiv’d paper, Alexander Terenin and David Draper devise a set of axioms that make the Cox-Jaynes system of conditional probability rigorous. Furthermore, they show that the complete set of Kolmogorov axioms (including countable additivity) can be derived as theorems from their axioms by conditioning on the entire sample space.

This is a deep and fundamental paper, which unfortunately means that I most probably do not grasp it’s complexities (especially as, for some reason, I keep reading it in pubs!). However I’m going to have a shot at having some thoughts on it, because I feel like it’s the sort of paper one should have thoughts on.