A bunch of ABC papers on arXiv yesterday, most of them linked to the incoming Handbook of ABC:
Apart from subjecting my [surprisingly large!] audience to three hours of ABC tutorial today, and after running Ponte della la Libertà to Mestre and back in a deep fog, I attended the second part of the 1st Italian-French statistics seminar at Ca’Foscari, Venetiarum Universitas, with talks by Stéfano Tonellato and Roberto Casarin. Stéfano discussed a most interesting if puzzling notion of clustering via Dirichlet process mixtures. Which indeed puzzles me for its dependence on the Dirichlet measure and on the potential for an unlimited number of clusters as the sample size increases. The method offers similarities with an approach from our 2000 JASA paper on running inference on mixtures without proper label switching, in that looking at pairs of allocated observations to clusters is revealing about the [true or pseudo-true] number of clusters. With divergence in using eigenvalues of Laplacians on similarity matrices. But because of the potential for the number of components to diverge I wonder at the robustness of the approach via non-parametric [Bayesian] modelling. Maybe my difficulty stands with the very notion of cluster, which I find poorly defined and mostly in the eyes of the beholder! And Roberto presented a recent work on SURE and VAR models, with a great graphical representation of the estimated connections between factors in a sparse graphical model.
While preparing crêpes at home yesterday night, I browsed through the most recent issue of Significance and among many goodies, I spotted an article by McKay and co-authors discussing the simulation of a British vs. German naval battle from the First World War I had never heard of, the Battle of the Dogger Bank. The article was illustrated by a few historical pictures, but I quickly came across a more statistical description of the problem, which was not about creating wargames and alternate realities but rather inferring about the likelihood of the actual income, i.e., whether or not the naval battle outcome [which could be seen as a British victory, ending up with 0 to 1 sunk boat] was either a lucky strike or to be expected. And the method behind solving this question was indeed both Bayesian and ABC-esque! I did not read the longer paper by McKay et al. (hard to do while flipping crêpes!) but the description in Significance was clear enough to understand that the six summary statistics used in this ABC implementation were the number of shots, hits, and lost turrets for both sides. 
(The answer to the original question is that indeed the British fleet was lucky to keep all its boats afloat. But it is also unlikely another score would have changed the outcome of WWI.) [As I found in this other history paper, ABC seems quite popular in historical inference! And there is another completely unrelated arXived paper with main title The Fog of War…]
The exact title of the paper by Jovana Metrovic, Dino Sejdinovic, and Yee Whye Teh is DR-ABC: Approximate Bayesian Computation with Kernel-Based Distribution Regression. It appeared last year in the proceedings of ICML. The idea is to build ABC summaries by way of reproducing kernel Hilbert spaces (RKHS). Regressing such embeddings to the “optimal” choice of summary statistics by kernel ridge regression. With a possibility to derive summary statistics for quantities of interest rather than for the entire parameter vector. The use of RKHS reminds me of Arthur Gretton’s approach to ABC, although I see no mention made of that work in the current paper.
In the RKHS pseudo-linear formulation, the prediction of a parameter value given a sample attached to this value looks like a ridge estimator in classical linear estimation. (I thus wonder at why one would stop at the ridge stage instead of getting the full Bayes treatment!) Things get a bit more involved in the case of parameters (and observations) of interest, as the modelling requires two RKHS, because of the conditioning on the nuisance observations. Or rather three RHKS. Since those involve a maximum mean discrepancy between probability distributions, which define in turn a sort of intrinsic norm, I also wonder at a Wasserstein version of this approach.
What I find hard to understand in the paper is how a large-dimension large-size sample can be managed by such methods with no visible loss of information and no explosion of the computing budget. The authors mention Fourier features, which never rings a bell for me, but I wonder how this operates in a general setting, i.e., outside the iid case. The examples do not seem to go into enough details for me to understand how this massive dimension reduction operates (and they remain at a moderate level in terms of numbers of parameters). I was hoping Jovana Mitrovic could present her work here at the 17w5025 workshop but she sadly could not make it to Banff for lack of funding!
The ABC workshop I co-organised has now started and, despite a few last minutes cancellations, we have gathered a great crowd of researchers on the validation and expansion of ABC methods. Or ABC’ory to keep up with my naming of workshops. The videos of the talks should come up progressively on the BIRS webpage. When I did not forget to launch the recording. The program is quite open and with this size of workshop allows for talks and discussions to last longer than planned: the first days contain several expository talks on ABC convergence, auxiliary or synthetic models, summary constructions, challenging applications, dynamic models, and model assessment. Plus prepared discussions on those topics that hopefully involve several workshop participants. We had also set some time for snap-talks, to induce everyone to give a quick presentation of one’s on-going research and open problems. The first day was rather full but saw a lot of interactions and discussions during and around the talks, a mood I hope will last till Friday! Today in replacement of Richard Everitt who alas got sick just before the workshop, we are conducting a discussion on dimensional issues, part of which is made of parts of the following slides (mostly recycled from earlier talks, including the mini-course in Les Diablerets):
Je reviendrai à Montréal, as the song by Robert Charlebois goes, for the MCM 2017 meeting there, on July 3-7. I was invited to give a plenary talk by the organisers of the conference . Along with
Steffen Dereich, WWU Münster, Germany
Paul Dupuis, Brown University, Providence, USA
Mark Girolami, Imperial College London, UK
Emmanuel Gobet, École Polytechnique, Palaiseau, France
Aicke Hinrichs, Johannes Kepler University, Linz, Austria
Alexander Keller, NVIDIA Research, Germany
Gunther Leobacher, Johannes Kepler University, Linz, Austria
Art B. Owen, Stanford University, USA
Note that, while special sessions are already selected, including oneon Stochastic Gradient methods for Monte Carlo and Variational Inference, organised by Victor Elvira and Ingmar Schuster (my only contribution to this session being the suggestion they organise it!), proposals for contributed talks will be selected based on one-page abstracts, to be submitted by March 1.
“…construction of low dimensional summary statistics can be performed as in a black box…”
Today Zhou and Fukuzumi just arXived a paper that proposes a gradient-based dimension reduction for ABC summary statistics, in the spirit of RKHS kernels as advocated, e.g., by Arthur Gretton. Here the projection is a mere linear projection Bs of the vector of summary statistics, s, where B is an estimated Hessian matrix associated with the posterior expectation E[θ|s]. (There is some connection with the latest version of Li’s and Fearnhead’s paper on ABC convergence as they also define a linear projection of the summary statistics, based on asymptotic arguments, although their matrix does depend on the true value of the parameter.) The linearity sounds like a strong restriction [to me] especially when the summary statistics have no reason to belong to a vectorial space and thus be open to changes of bases and linear projections. For instance, a specific value taken by a summary statistic, like 0 say, may be more relevant than the range of their values. On a larger scale, I am doubtful about always projecting a vector of summary statistics on a subspace with the smallest possible dimension, ie the dimension of θ. In practical settings, it seems impossible to derive the optimal projection and a subvector is almost certain to loose information against a larger vector.
“Another proposal is to use different summary statistics for different parameters.”
Which is exactly what we did in our random forest estimation paper. Using a different forest for each parameter of interest (but no real tree was damaged in the experiment!).
