noisy importance sampling

A recent short arXival by Fernando Llorente, Luca Martino, Jesse Read, and David Delgado–Gómez in which they analyse settings where (only) a noisy version of the target density is available. Not necessarily in an unbiased fashion although the paper is somewhat unclear as to which integral is targeted in (6), since the integrand is not the original target p(x). The following development is about finding the optimal importance function, which differs from the usual due to the random nature of the approximation, but it does not seem to reconnect with the true target p(x), except when the noisy realisation is unbiased… To me this is a major issue in simulation methodology in that getting away from the unbiasedness constraint opens (rather obviously) a much wider choice of techniques.

MCMC importance samplers for intractable likelihoods

Jordan Franks just posted on arXiv his PhD dissertation at the University of Jyväskylä, where he discuses several of his works:

1. M. Vihola, J. Helske, and J. Franks. Importance sampling type estimators based on approximate marginal MCMC. Preprint arXiv:1609.02541v5, 2016.
2. J. Franks and M. Vihola. Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance. Preprint arXiv:1706.09873v4, 2017.
3. J. Franks, A. Jasra, K. J. H. Law and M. Vihola.Unbiased inference for discretely observed hidden Markov model diffusions. Preprint arXiv:1807.10259v4, 2018.
4. M. Vihola and J. Franks. On the use of ABC-MCMC with inflated tolerance and post-correction. Preprint arXiv:1902.00412, 2019

focusing on accelerated approximate MCMC (in the sense of pseudo-marginal MCMC) and delayed acceptance (as in our recently accepted paper). Comparing delayed acceptance with MCMC importance sampling to the advantage of the later. And discussing the choice of the tolerance sequence for ABC-MCMC. (Although I did not get from the thesis itself the target of the improvement discussed.)

approximate likelihood perspective on ABC

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.

simulated summary statistics [in the sky]

Thinking it was related with ABC, although in the end it is not!, I recently read a baffling cosmology paper by Jeffrey and Abdalla. The data d there means an observed (summary) statistic, while the summary statistic is a transform of the parameter, μ(θ), which calibrates the distribution of the data. With nuisance parameters. More intriguing to me is the sentence that the correct likelihood of d is indexed by a simulated version of μ(θ), μ'(θ), rather than by μ(θ). Which seems to assume that the pseudo- or simulated data can be produced for the same value of the parameter as the observed data. The rest of the paper remains incomprehensible for I do not understand how the simulated versions are simulated.

“…the corrected likelihood is more than a factor of exp(30) more probable than the uncorrected. This is further validation of the corrected likelihood; the model (i.e. the corrected likelihood) shows a better goodness-of-fit.”

The authors further ressort to Bayes factors to compare corrected and uncorrected versions of the likelihoods, which leads (see quote) to picking the corrected version. But are they comparable as such, given that the corrected version involves simulations that are treated as supplementary data? As noted by the authors, the Bayes factor  unsurprisingly goes to one as the number M of simulations grows to infinity, as supported by the graph below.

ISBA 2016 [#3]

Among the sessions I attended yesterday, I really liked the one on robustness and model mispecification. Especially the talk by Steve McEachern on Bayesian inference based on insufficient statistics, with a striking graph of the degradation of the Bayes factor as the prior variance increases. I sadly had no time to grab a picture of the graph, which compared this poor performance against a stable rendering when using a proper summary statistic. It clearly relates to our work on ABC model choice, as well as to my worries about the Bayes factor, so this explains why I am quite excited about this notion of restricted inference. In this session, Chris Holmes also summarised his two recent papers on loss-based inference, which I discussed here in a few posts, including the Statistical Science discussion Judith and I wrote recently. I also went to the j-ISBA [section] session which was sadly under-attended, maybe due to too many parallel sessions, maybe due to the lack of unifying statistical theme.