ABC in Clermont-Ferrand

Today I am taking part in a one-day workshop at the Université of Clermont Auvergne on ABC. With applications to cosmostatistics, along with Martin Kilbinger [with whom I worked on PMC schemes], Florent Leclerc and Grégoire Aufort. This should prove a most exciting day! (With not enough time to run up Puy de Dôme in the morning, though.)

unimaginable scale culling

Despite the evidence brought by ABC on the inefficiency of culling in massive proportions the British Isles badger population against bovine tuberculosis, the [sorry excuse for a] United Kingdom government has permitted a massive expansion of badger culling, with up to 64,000 animals likely to be killed this autumn… Since the cows are the primary vectors of the disease, what about starting with these captive animals?!

a problem that did not need ABC in the end

While in Denver, at JSM, I came across [across validated!] this primarily challenging problem of finding the posterior of the 10³ long probability vector of a Multinomial M(10⁶,p) when only observing the range of a realisation of M(10⁶,p). This sounded challenging because the distribution of the pair (min,max) is not available in closed form. (Although this allowed me to find a paper on the topic by the late Shanti Gupta, who was chair at Purdue University when I visited 32 years ago…) This seemed to call for ABC (especially since I was about to give an introductory lecture on the topic!, law of the hammer…), but the simulation of datasets compatible with the extreme values of both minimum and maximum, m=80 and M=12000, proved difficult when using a uniform Dirichlet prior on the probability vector, since these extremes called for both small and large values of the probabilities. However, I later realised that the problem could be brought down to a Multinomial with only three categories and the observation (m,M,n-m-M), leading to an obvious Dirichlet posterior and a predictive for the remaining 10³-2 realisations.

Introductory overview lecture: the ABC of ABC [JSM19 #1]

Here are my slides [more or less] for the introductory overview lecture I am giving today at JSM 2019, 4:00-5:50, CC-Four Seasons I. There is obviously quite an overlap with earlier courses I gave on the topic, although I refrained here from mentioning any specific application (like population genetics) to focus on statistical and computational aspects.

Along with the other introductory overview lectures in this edition of JSM:

• Sunday 28, 2:00-3:50, CC-Four Seasons I: CSI at the JSM: Forensic Statistics and the Value of Scientific Evidence in Court by Hal Stern (University of California, Irvine)
• Monday 29, 8:30-10:20, CC-205: Assessing Procedures vs. Assessing Evidence by Michael Levine (University of Massachusetts, Amherst)
• Monday 29, 2:00_3:50, CC-205: Causal inference in modern statistics by Jennifer Hill (New York University)and Avi Feller (UC Berkeley)
• Tuesday 30, 8:30-10:20, CC-205: Modern Risk Analysis  by Walter Piergorsch (University of Arizona) and David Banks (Duke University)

ABC with Gibbs steps

With Grégoire Clarté, Robin Ryder and Julien Stoehr, all from Paris-Dauphine, we have just arXived a paper on the specifics of ABC-Gibbs, which is a version of ABC where the generic ABC accept-reject step is replaced by a sequence of n conditional ABC accept-reject steps, each aiming at an ABC version of a conditional distribution extracted from the joint and intractable target. Hence an ABC version of the standard Gibbs sampler. What makes it so special is that each conditional can (and should) be conditioning on a different statistic in order to decrease the dimension of this statistic, ideally down to the dimension of the corresponding component of the parameter. This successfully bypasses the curse of dimensionality but immediately meets with two difficulties. The first one is that the resulting sequence of conditionals is not coherent, since it is not a Gibbs sampler on the ABC target. The conditionals are thus incompatible and therefore convergence of the associated Markov chain becomes an issue. We produce sufficient conditions for the Gibbs sampler to converge to a stationary distribution using incompatible conditionals. The second problem is then that, provided it exists, the limiting and also intractable distribution does not enjoy a Bayesian interpretation, hence may fail to be justified from an inferential viewpoint. We however succeed in producing a version of ABC-Gibbs in a hierarchical model where the limiting distribution can be explicited and even better can be weighted towards recovering the original target. (At least with limiting zero tolerance.)

the true meaning of ABC

holistic framework for ABC

An AISTATS 2019 paper was recently arXived by Kelvin Hsu and Fabio Ramos. Proposing an ABC method

“…consisting of (1) a consistent surrogate likelihood model that modularizes queries from simulation calls, (2) a Bayesian learning objective for hyperparameters that improves inference accuracy, and (3) a posterior surrogate density and a super-sampling inference algorithm using its closed-form posterior mean embedding.”

While this sales line sounds rather obscure to me, the authors further defend their approach against ABC-MCMC or synthetic likelihood by the points

“that (1) only one new simulation is required at each new parameter θ and (2) likelihood queries do not need to be at parameters where simulations are available.”

using a RKHS approach to approximate the likelihood or the distribution of the summary (statistic) given the parameter (value) θ. Based on the choice of a certain positive definite kernel. (As usual, I do not understand why RKHS would do better than another non-parametric approach, especially since the approach approximates the full likelihood, but I am not a non-parametrician…)

“The main advantage of using an approximate surrogate likelihood surrogate model is that it readily provides a marginal surrogate likelihood quantity that lends itself to a hyper-parameter learning algorithm”

The tolerance ε (and other cyberparameters) are estimated by maximising the approximated marginal likelihood, which happens to be available in the convenient case the prior is an anisotropic Gaussian distribution. For the simulated data in the reference table? But then missing the need for localising the simulations near the posterior? Inference is then conducting by simulating from this approximation. With the common (to RKHS) drawback that the approximation is “bounded and normalized but potentially non-positive”.