## variational approximation to empirical likelihood ABC

Posted in Statistics with tags , , , , , , , , , , , , , , , , , , on October 1, 2021 by xi'an

Sanjay Chaudhuri and his colleagues from Singapore arXived last year a paper on a novel version of empirical likelihood ABC that I hadn’t yet found time to read. This proposal connects with our own, published with Kerrie Mengersen and Pierre Pudlo in 2013 in PNAS. It is presented as an attempt at approximating the posterior distribution based on a vector of (summary) statistics, the variational approximation (or information projection) appearing in the construction of the sampling distribution of the observed summary. (Along with a weird eyed-g symbol! I checked inside the original LaTeX file and it happens to be a mathbbmtt g, that is, the typewriter version of a blackboard computer modern g…) Which writes as an entropic correction of the true posterior distribution (in Theorem 1).

“First, the true log-joint density of the observed summary, the summaries of the i.i.d. replicates and the parameter have to be estimated. Second, we need to estimate the expectation of the above log-joint density with respect to the distribution of the data generating process. Finally, the differential entropy of the data generating density needs to be estimated from the m replicates…”

The density of the observed summary is estimated by empirical likelihood, but I do not understand the reasoning behind the moment condition used in this empirical likelihood. Indeed the moment made of the difference between the observed summaries and the observed ones is zero iff the true value of the parameter is used in the simulation. I also fail to understand the connection with our SAME procedure (Doucet, Godsill & X, 2002), in that the empirical likelihood is based on a sample made of pairs (observed,generated) where the observed part is repeated m times, indeed, but not with the intent of approximating a marginal likelihood estimator… The notion of using the actual data instead of the true expectation (i.e. as a unbiased estimator) at the true parameter value is appealing as it avoids specifying the exact (or analytical) value of this expectation (as in our approach), but I am missing the justification for the extension to any parameter value. Unless one uses an ancillary statistic, which does not sound pertinent… The differential entropy is estimated by a Kozachenko-Leonenko estimator implying k-nearest neighbours.

“The proposed empirical likelihood estimates weights by matching the moments of g(X¹), , g(X) with that of
g(X), without requiring a direct relationship with the parameter. (…) the constraints used in the construction of the empirical likelihood are based on the identity in (7), which can only be satisfied when θ = θ⁰. “

Although I am feeling like missing one argument, the later part of the paper seems to comfort my impression, as quoted above. Meaning that the approximation will fare well only in the vicinity of the true parameter. Which makes it untrustworthy for model choice purposes, I believe. (The paper uses the g-and-k benchmark without exploiting Pierre Jacob’s package that allows for exact MCMC implementation.)

## approximate likelihood perspective on ABC

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , on December 20, 2018 by xi'an

George Karabatsos and Fabrizio Leisen have recently published in Statistics Surveys a fairly complete survey on ABC methods [which earlier arXival I had missed]. Listing within an extensive bibliography of 20 pages some twenty-plus earlier reviews on ABC (with further ones in applied domains)!

“(…) any ABC method (algorithm) can be categorized as either (1) rejection-, (2) kernel-, and (3) coupled ABC; and (4) synthetic-, (5) empirical- and (6) bootstrap-likelihood methods; and can be combined with classical MC or VI algorithms [and] all 22 reviews of ABC methods have covered rejection and kernel ABC methods, but only three covered synthetic likelihood, one reviewed the empirical likelihood, and none have reviewed coupled ABC and bootstrap likelihood methods.”

The motivation for using approximate likelihood methods is provided by the examples of g-and-k distributions, although the likelihood can be efficiently derived by numerical means, as shown by Pierre Jacob‘s winference package, of mixed effect linear models, although a completion by the mixed effects themselves is available for Gibbs sampling as in Zeger and Karim (1991), and of the hidden Potts model, which we covered by pre-processing in our 2015 paper with Matt Moores, Chris Drovandi, Kerrie Mengersen. The paper produces a general representation of the approximate likelihood that covers the algorithms listed above as through the table below (where t(.) denotes the summary statistic):

The table looks a wee bit challenging simply because the review includes the synthetic likelihood approach of Wood (2010), which figured preeminently in the 2012 Read Paper discussion but opens the door to all kinds of approximations of the likelihood function, including variational Bayes and non-parametric versions. After a description of the above versions (including a rather ignored coupled version) and the special issue of ABC model choice,  the authors expand on the difficulties with running ABC, from multiple tuning issues, to the genuine curse of dimensionality in the parameter (with unnecessary remarks on low-dimension sufficient statistics since they are almost surely inexistent in most realistic settings), to the mis-specified case (on which we are currently working with David Frazier and Judith Rousseau). To conclude, an worthwhile update on ABC and on the side a funny typo from the reference list!

Li, W. and Fearnhead, P. (2018, in press). On the asymptotic efficiency
of approximate Bayesian computation estimators. Biometrika na na-na.