a generalized representation of Bayesian inference

Jeremias Knoblauch, Jack Jewson and Theodoros Damoulas, all affiliated with Warwick (hence a potentially biased reading!), arXived a paper on loss-based Bayesian inference that Jack discussed with me on my last visit to Warwick. As I was somewhat scared by the 61 pages, of which the 8 first pages are in NeurIPS style. The authors argue for a decision-theoretic approach to Bayesian inference that involves a loss over distributions and a divergence from the prior. For instance, when using the log-score as the loss and the Kullback-Leibler divergence, the regular posterior emerges, as shown by Arnold Zellner. Variational inference also falls under this hat. The argument for this generalization is that any form of loss can be used and still returns a distribution that is used to assess uncertainty about the parameter (of interest). In the axioms they produce for justifying the derivation of the optimal procedure, including cases where the posterior is restricted to a certain class, one [Axiom 4] generalizes the likelihood principle. Given the freedom brought by this general framework, plenty of fringe Bayes methods like standard variational Bayes can be seen as solutions to such a decision problem. Others like EP do not. Of interest to me are the potentials for this formal framework to encompass misspecification and likelihood-free settings, as well as for assessing priors, which is always a fishy issue. (The authors mention in addition the capacity to build related specific design Bayesian deep networks, of which I know nothing.) The obvious reaction of mine is one of facing an abundance of wealth (!) but encompassing approximate Bayesian solutions within a Bayesian framework remains an exciting prospect.

nonparametric Bayesian clay for robust decision bricks

Just received an email today that our discussion with Judith of Chris Holmes and James Watson’s paper was now published as Statistical Science 2016, Vol. 31, No. 4, 506-510… While it is almost identical to the arXiv version, it can be read on-line.

non-identifiability in Venezia

Last Wednesday, I attended a seminar by T. Kitagawa at the economics seminar of the University Ca’ Foscari, in Venice, which was about (uncertain) identifiability and a sort of meta-Bayesian approach to the problem. Just to give an intuition about the setting, a toy example is a simultaneous equation model Ax=ξ, where x and ξ are two-dimensional vectors, ξ being a standard bivariate Normal noise. In that case, A is not completely identifiable. The argument in the talk (and the paper) is that the common Bayesian answer that sets a prior on the non-identifiable part (which is an orthogonal matrix in the current setting) is debatable as it impacts inference on the non-identifiable parts, even in the long run. Which seems fine from my viewpoint. The authors propose to instead consider the range of possible priors that are compatible with the set restrictions on the non-identifiable parts and to introduce a mixture between a regular prior on the whole parameter A and this collection of priors, which can be seen as a set-valued prior although this does not fit within the Bayesian framework in my opinion. Once this mixture is constructed, a formal posterior weight on the regular prior can be derived. As well as a range of posterior values for all quantities of interest. While this approach connects with imprecise probabilities à la Walley (?) and links with robust Bayesian studies of the 1980’s, I always have difficulties with the global setting of such models, which do not come under criticism while being inadequate. (Of course, there are many more things I do not understand in econometrics!)

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!)