**A** great Bayesian Analysis webinar this afternoon with well-balanced presentations by Steve MacEachern and John Lewis, and original discussions by Bertrand Clarke and Fabrizio Rugieri. Which attracted 122 participants. I particularly enjoyed Bertrand’s points that likelihoods were more general than models [made in 6 different wordings!] and that this paper was closer to the M-open perspective. I think I eventually got the reason why the approach could be seen as an ABC with ε=0, since the simulated y’s all get the right statistic, but this presentation does not bring a strong argument in favour of the restricted likelihood approach, when considering the methodological and computational effort. The discussion also made me wonder if tools like VAEs could be used towards approximating the distribution of T(y) conditional on the parameter θ. This is also an opportunity to thank my friend Michele Guindani for his hard work as Editor of Bayesian Analysis and in particular for keeping the discussion tradition thriving!

## Archive for discussion

## BA webinar with discussion

Posted in Statistics with tags #BAjournal, ABC, Bayesian Analysis, conditional sampling, discussion, insufficiency, MCMC, restricted likelihood, webinar on February 8, 2022 by xi'an## likelihood-free inference by ratio estimation

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags ABC, Austria, Bayesian lasso, Charlie Geyer, conflict of interest, Cross Validation, discussion, estimating constants, likelihood-free methods, logistic regression, nonparametrics, normalising constant, Salzburg, SimStat2019, summer, University of Warwick on September 9, 2019 by xi'an

“This approach for posterior estimation with generative models mirrors the approach of Gutmann and Hyvärinen (2012) for the estimation of unnormalised models. The main difference is that here we classify between two simulated data sets while Gutmann and Hyvärinen (2012) classified between the observed data and simulated reference data.”

**A** 2018 arXiv posting by Owen Thomas et al. (including my colleague at Warwick, Rito Dutta, *CoI warning!*) about estimating the likelihood (and the posterior) when it is intractable. Likelihood-free but not ABC, since the ratio likelihood to marginal is estimated in a non- or semi-parametric (and biased) way. Following Geyer’s 1994 fabulous estimate of an unknown normalising constant via logistic regression, the current paper which I read in preparation for my discussion in the ABC optimal design in Salzburg uses probabilistic classification and an exponential family representation of the ratio. Opposing data from the density and data from the marginal, assuming both can be readily produced. The logistic regression minimizing the asymptotic classification error is the logistic transform of the log-ratio. For a finite (double) sample, this minimization thus leads to an empirical version of the ratio. Or to a smooth version if the log-ratio is represented as a convex combination of summary statistics, turning the approximation into an exponential family, which is a clever way to buckle the buckle towards ABC notions. And synthetic likelihood. Although with a difference in estimating the exponential family parameters β(θ) by minimizing the classification error, parameters that are indeed conditional on the parameter θ. Actually the paper introduces a further penalisation or regularisation term on those parameters β(θ), which could have been processed by Bayesian Lasso instead. This step is essentially dirving the selection of the summaries, except that it is for each value of the parameter θ, at the expense of a X-validation step. This is quite an original approach, as far as I can tell, but I wonder at the link with more standard density estimation methods, in particular in terms of the precision of the resulting estimate (and the speed of convergence with the sample size, if convergence there is).

## discussione a Padova

Posted in Statistics, University life with tags ABC, ABC model choice, Biometrika, calibration, credible set, discussion, empirical Bayes methods, empirical likelihood, Italia, Madrid, Padova, Trieste, Università degli studi di Padova on March 25, 2013 by xi'an**H**ere are the slides of my talk in Padova for the workshop Recent Advances in statistical inference: theory and case studies (very similar to the slides for the Varanasi and Gainesville meetings, obviously!, with Peter Müller commenting [at last!] that I had picked the wrong photos from Khajuraho!)

**T**he worthy Padova addendum is that I had two discussants, Stefano Cabras from Universidad Carlos III in Madrid, whose slides are :

and Francesco Pauli, from Trieste, whose slides are:

**T**hese were kind and rich discussions with many interesting openings: Stefano’s idea of estimating the pivotal function *h* is opening new directions, obviously, as it indicates an additional degree of freedom in calibrating the method. Esp. when considering the high variability of the empirical likelihood fit depending on the the function *h*. For instance, one could start with a large collection of candidate functions and build a regression or a principal component reparameterisation from this collection… (Actually I did not get point #1 about ignoring *f*: the empirical likelihood is by essence ignoring anything outside the identifying equation, so as long as the equation is valid..) Point #2: Opposing sample free and simulation free techniques is another interesting venue, although I would not say ABC is “sample free”. As to point #3, I will certainly get a look at Monahan and Boos (1992) to see if this can drive the choice of a specific type of pseudo-likelihoods. I like the idea of checking the “coverage of posterior sets” and even more “the likelihood must be the density of a statistic, not necessarily sufficient” as it obviously relates with our current ABC model comparison work… Esp. when the very same paper is mentioned by Francesco as well. ** Grazie, Stefano!** I also appreciate the survey made by Francesco on the consistency conditions, because I think this is an important issue that should be taken into consideration when designing ABC algorithms. (Just pointing out again that, in the theorem of Fearnhead and Prangle (2012) quoting Bernardo and Smith (1992), some conditions are missing for the mathematical consistency to apply.) I also like the agreement we seem to reach about ABC being evaluated per se rather than an a poor man’s Bayesian method. Francesco’s analysis of Monahan and Boos (1992) as validating or not empirical likelihood points out a possible link with the recent coverage analysis of Prangle et al., discussed on the ‘Og a few weeks ago. And an unsuspected link with Larry Wasserman!

*Grazie, Francesco!*