**J**ust found out that our paper On parameter estimation with the Wasserstein distance with Espen Bernton, Pierre Jacob, and Mathieu Gerber, has now appeared on-line on Information and Inference: A Journal of the IMA,

## Archive for misspecified model

## on-line parameter estimation with Wasserstein

Posted in Books, Statistics, University life with tags ABC, consistency of ABC methods, empirical distribution, IMA, Information and Inference, misspecified model, Oxford University Press, publication, Wasserstein distance on November 27, 2019 by xi'an## uncertainty in the ABC posterior

Posted in Statistics with tags ABC, Bayesian Analysis, Gaussian processes, misspecified model, Monte Carlo error, prior predictive, synthetic likelihood on July 24, 2019 by xi'an**I**n the most recent Bayesian Analysis, Marko Järvenpää et al. (including my coauthor Aki Vehtari) consider an ABC setting where the number of available simulations of pseudo-samples is limited. And where they want to quantify the amount of uncertainty resulting from the estimation of the ABC posterior density. Which is a version of the Monte Carlo error in practical ABC, in that this is the difference between the ABC posterior density for a given choice of summaries and a given choice of tolerance, and the actual approximation based on a finite number of simulations from the prior predictive. As in earlier works by Michael Gutmann and co-authors, the focus stands in designing a sequential strategy to decide where to sample the next parameter value towards minimising a certain expected loss. And in adopting a Gaussian process modelling for the discrepancy between observed data and simulated data, hence generalising the synthetic likelihood approach. This allows them to compute the expectation and the variance of the unnormalised ABC posterior, based on plugged-in estimators. From where the authors derive a loss as the expected variance of the acceptance probability (although it is not parameterisation invariant). I am unsure I see the point for this choice in that there is no clear reason for the resulting sequence of parameter choices to explore the support of the posterior distribution in a relatively exhaustive manner. The paper also mentions alternatives where the next parameter is chosen at the location where “the uncertainty of the unnormalised ABC posterior is highest”. Which sounds more pertinent to me. And further avoids integrating out the parameter. I also wonder if ABC mis-specification analysis could apply in this framework since the Gaussian process is most certainly a “wrong” model. (When concluding this post, I realised I had written a similar entry two years ago about the earlier version of the paper!)

## postdoc position still open

Posted in pictures, Statistics, University life with tags ABC, Agence Nationale de la Recherche, ANR, approximate Bayesian inference, bois de Boulogne, La Défense, misspecified model, Paris, Paris-Saclay campus, PhD thesis, postdoctoral position, PSL Research University, Université de Montpellier, Université Paris Dauphine, University of Oxford on May 30, 2019 by xi'an**T**he post-doctoral position supported by the ANR funding of our Paris-Saclay-Montpellier research conglomerate on approximate Bayesian inference and computation remains open for the time being. We are more particularly looking for candidates with a strong background in mathematical statistics, esp. Bayesian non-parametrics, towards the analysis of the limiting behaviour of approximate Bayesian inference. Candidates should email me (gmail address: bayesianstatistics) with a detailed vita (CV) and a motivation letter including a research plan. Letters of recommendation may also be emailed to the same address.

## robust Bayesian synthetic likelihood

Posted in Statistics with tags ABC, Australia, Bayesian synthetic likelihood, Brisbane, industrial ruins, MCMC, Melbourne, Metropolis-Hastings algorithm, misspecified model, Monash University, pseudo-likelihood, QUT, summary statistics, Sydney Harbour on May 16, 2019 by xi'an**D**avid Frazier (Monash University) and Chris Drovandi (QUT) have recently come up with a robustness study of Bayesian synthetic likelihood that somehow mirrors our own work with David. In a sense, Bayesian synthetic likelihood is definitely misspecified from the start in assuming a Normal distribution on the summary statistics. When the data generating process is misspecified, even were the Normal distribution the “true” model or an appropriately converging pseudo-likelihood, the simulation based evaluation of the first two moments of the Normal is biased. Of course, for a choice of a summary statistic with limited information, the model can still be *weakly compatible* with the data in that there exists a pseudo-true value of the parameter θ⁰ for which the synthetic mean μ(θ⁰) is the mean of the statistics. (Sorry if this explanation of mine sounds unclear!) Or rather the Monte Carlo estimate of μ(θ⁰) coincidences with that mean.The same Normal toy example as in our paper leads to very poor performances in the MCMC exploration of the (unsympathetic) synthetic target. The robustification of the approach as proposed in the paper is to bring in an extra parameter to correct for the bias in the mean, using an additional Laplace prior on the bias to aim at sparsity. Or the same for the variance matrix towards inflating it. This over-parameterisation of the model obviously avoids the MCMC to get stuck (when implementing a random walk Metropolis with the target as a scale).

## did variational Bayes work?

Posted in Books, Statistics with tags approximate Bayesian inference, asymptotic Bayesian methods, ICML 2018, importance sampling, misspecified model, Pareto distribution, Pareto smoothed importance sampling, posterior predictive, variational Bayes methods, what you get is what you see on May 2, 2019 by xi'an**A**n interesting ICML 2018 paper by Yuling Yao, Aki Vehtari, Daniel Simpson, and Andrew Gelman I missed last summer on [the fairly important issue of] assessing the quality or lack thereof of a variational Bayes approximation. In the sense of being near enough from the true posterior. The criterion that they propose in this paper relates to the Pareto smoothed importance sampling technique discussed in an earlier post and which I remember discussing with Andrew when he visited CREST a few years ago. The truncation of the importance weights of prior x likelihood / VB approximation avoids infinite variance issues but induces an unknown amount of bias. The resulting diagnostic is based on the estimation of the Pareto order k. If the true value of k is less than ½, the variance of the associated Pareto distribution is finite. The paper suggests to conclude at the worth of the variational approximation when the estimate of k is less than 0.7, based on the empirical assessment of the earlier paper. The paper also contains a remark on the poor performances of the generalisation of this method to marginal settings, that is, when the importance weight is the ratio of the true and variational marginals for a sub-vector of interest. I find the counter-performances somewhat worrying in that Rao-Blackwellisation arguments make me prefer marginal ratios to joint ratios. It may however be due to a poor approximation of the marginal ratio that reflects on the approximation and not on the ratio itself. A second proposal in the paper focus on solely the point estimate returned by the variational Bayes approximation. Testing that the posterior predictive is well-calibrated. This is less appealing, especially when the authors point out the “dissadvantage is that this diagnostic does not cover the case where the observed data is not well represented by the model.” In other words, misspecified situations. This potential misspecification could presumably be tested by comparing the Pareto fit based on the actual data with a Pareto fit based on simulated data. Among other deficiencies, they point that this is “a local diagnostic that will not detect unseen modes”. In other words, *what you get is what you see*.

## absint[he] post-doc on approximate Bayesian inference in Paris, Montpellier and Oxford

Posted in Statistics with tags ABC, Agence Nationale de la Recherche, ANR, approximate Bayesian inference, bois de Boulogne, La Défense, misspecified model, Paris, Paris-Saclay campus, PhD thesis, postdoctoral position, Université de Montpellier, Université Paris Dauphine, University of Oxford on March 18, 2019 by xi'anAs a consequence of its funding by the Agence Nationale de la Recherche (ANR) in 2018, the ABSint research conglomerate is now actively recruiting a post-doctoral collaborator for up to 24 months. The accronym ** ABSint** stands for Approximate Bayesian solutions for inference on large datasets and complex models. The ABSint conglomerate involves researchers located in Paris, Saclay, Montpelliers, as well as Lyon, Marseille, Nice. This call seeks candidates with an excellent research record and who are interested to collaborate with local researchers on approximate Bayesian techniques like ABC, variational Bayes, PAC-Bayes, Bayesian non-parametrics, scalable MCMC, and related topics. A potential direction of research would be the derivation of new Bayesian tools for model checking in such complex environments. The post-doctoral collaborator will be primarily located in Université Paris-Dauphine, with supported periods in Oxford and visits to Montpellier. No teaching duty is attached to this research position.

Applications can be submitted in either English or French. Sufficient working fluency in English is required. While mastering some French does help with daily life in France (!), it is not a prerequisite. The candidate must hold a PhD degree by the date of application (not the date of employment). Position opens on July 01, with possible accommodation for a later start in September or October.

Deadline for application is April 30 or until position filled. Estimated gross salary is around 2500 EUR, depending on experience (years) since PhD. Candidates should contact Christian Robert (gmail address: bayesianstatistics) with a detailed vita (CV) and a motivation letter including a research plan. Letters of recommendation may also be emailed to the same address.