Approximate Integrated Likelihood via ABC methods

My PhD student Clara Grazian just arXived this joint work with Brunero Liseo on using ABC for marginal density estimation. The idea in this paper is to produce an integrated likelihood approximation in intractable problems via the ratio

both terms in the ratio being estimated from simulations,

(with possible closed form for the denominator). Although most of the examples processed in the paper (Poisson means ratio, Neyman-Scott’s problem, g-&-k quantile distribution, semi-parametric regression) rely on summary statistics, hence de facto replacing the numerator above with a pseudo-posterior conditional on those summaries, the approximation remains accurate (for those examples). In the g-&-k quantile example, Clara and Brunero compare our ABC-MCMC algorithm with the one of Allingham et al. (2009, Statistics & Computing): the later does better by not replicating values in the Markov chain but instead proposing a new value until it is accepted by the usual Metropolis step. (Although I did not spend much time on this issue, I cannot see how both approaches could be simultaneously correct. Even though the outcomes do not look very different.) As noted by the authors, “the main drawback of the present approach is that it requires the use of proper priors”, unless the marginalisation of the prior can be done analytically. (This is an interesting computational problem: how to provide an efficient approximation to a marginal density of a σ-finite measure, assuming this density exists.)

Clara will give a talk at CREST-ENSAE today about this work, in the Bayes in Paris seminar: 2pm in room 18.

ABC with indirect summary statistics

After reading Drovandi’s and Pettitt’s Bayesian Indirect Inference, I checked (in the plane to Birmingham) the earlier Gleim’s and Pigorsch’s Approximate Bayesian Computation with indirect summary statistics. The setting is indeed quite similar to the above, with a description of three ways of connecting indirect inference with ABC, albeit with a different range of illustrations. This preprint states most clearly its assumption that the generating model is a particular case of the auxiliary model, which sounds anticlimactic since the auxiliary model is precisely used because the original one is mostly out of reach! This certainly was the original motivation for using indirect inference.

The part of the paper that I find the most intriguing is the argument that the indirect approach leads to sufficient summary statistics, in the sense that they “are sufficient for the parameters of the auxiliary model and (…) sufficiency carries over to the model of interest” (p.31). Looking at the details in the Appendix, I found that the argument is lacking, because the likelihood as a functional is shown to be a (sufficient) statistic, which seems both a tautology and irrelevant because this is different from the likelihood considered at the (auxiliary) MLE, which is the summary statistic used in fine.

“…we expand the square root of an innovation density h in a Hermite expansion and truncate the infinite polynomial at some integer K which, together with other tuning parameters of the SNP density, has to be determined through a model selection criterion (such as BIC). Now we take the leading term of the Hermite expansion to follow a Gaussian GARCH model.”

As in Drovandi and Pettitt, the performances of the ABC-I schemes are tested on a toy example, which is a very basic exponential iid sample with a conjugate prior. With a gamma model as auxiliary. The authors use a standard ABC based on the first two moments as their benchmark, however they do not calibrate those moments in the distance and end up with poor performances of ABC (in a setting where there is a sufficient statistic!). The best choice in this experiment appears as the solution based on the score, but the variances of the distances are not included in the comparison tables. The second implementation considered in the paper is a rather daunting continuous-time non-Gaussian Ornstein-Uhlenbeck stochastic volatility model à la Barndorf-Nielsen and Shephard (2001). The construction of the semi-nonparametric (why not semi-parametric?) auxiliary model is quite involved as well, as illustrated by the quote above. The approach provides an answer, with posterior ABC-IS distributions on all parameters of the original model, which kindles the question of the validation of this answer in terms of the original posterior. Handling simultaneously several approximation processes would help in this regard.

Hidden Markov mixtures of regression

It took the RSS feed of Bayesian Analysis to disappear from my screen—because the Bayesian Analysis 4(4) issue was completed—for me to spot this very nice paper by Matthew A. Taddy and Athanasios Kottas on Markov switching regression models. It reminds me of earlier papers of mine’s with Monica Billio and Alain Monfort, and with Merrilee Hurn and Ana Justel, on Markov switching and mixtures of regression, respectively. At that time, with Merrilee, we had in mind to extend mixtures of regressions to generalised linear mixtures of generalised linear models but never found the opportunity to concretise the model. The current paper goes much farther by using mixtures of Dirichlet priors, thus giving a semi-parametric flavour to the mixture of regressions. There is also an interesting application to fishery management.

This issue also includes an emotional postnote by Brad Carlin, who is now stepping down from being the Bayesian Analysis Editor-in-chief. Brad unreservedly deserves thanks for mentoring Bayesian Analysis towards a wider audience and a stronger requirement on the papers being published in the journal. I think Bayesian Analysis now is a mainstream journal rather than the emanation of a society, albeit as exciting as ISBA! The electronic format adopted by Bayesian Analysis should be exploited further towards forums and on-line discussions of all papers, rather than singling out one paper by issue, and I am glad Brad agrees on this possible change of editorial policy. All the best to the new Editor-in-chief, Herbie Lee!