**T**his week, I gave a short and introductory course in Warwick for the CDT (PhD) students on my perceived connections between reverse logistic regression à la Geyer and GANS, among other things. The first attempt was cancelled in 2020 due to the pandemic, the second one in 2021 was on-line and thus offered little possibilities for interactions. Preparing for this third attempt made me read more papers on some statistical analyses of GANs and WGANs, which was more satisfactory [for me] even though I could not get into the technical details…

## Archive for Wasserstein distance

## Concentration and robustness of discrepancy-based ABC [One World ABC ‘minar, 28 April]

Posted in Statistics, University life with tags ABC, Approximate Bayesian computation, approximate Bayesian inference, discrepancy, Japan, One World ABC Seminar, RIKEN, sufficiency, Tokyo, University of Warwick, Wasserstein distance, webinar on April 15, 2022 by xi'an**O**ur next speaker at the One World ABC Seminar will be Pierre Alquier, who will talk about “

*Concentration and robustness of discrepancy-based ABC*“, on

**Thursday April 28, at 9.30am UK time,**with an abstract reported below.

Approximate Bayesian Computation (ABC) typically employs summary statistics to measure the discrepancy among the observed data and the synthetic data generated from each proposed value of the parameter of interest. However, finding good summary statistics (that are close to sufficiency) is non-trivial for most of the models for which ABC is needed. In this paper, we investigate the properties of ABC based on integral probability semi-metrics, including MMD and Wasserstein distances. We exhibit conditions ensuring the contraction of the approximate posterior. Moreover, we prove that MMD with an adequate kernel leads to very strong robustness properties.

## ABC for COVID spread reconstruction

Posted in Books, pictures, Statistics, Travel with tags ABC, airport, COVID-19, INSERM, Nature, pandemic, PLoS ONE, Voronoi tesselation, Wasserstein distance, wikipedia on December 27, 2021 by xi'an**A** recent Nature paper by Jessica Davis et al. (with an assessment by Simon Cauchemez and X from INSERM) reassessed the appearance of COVID in European and American States. Accounting for the massive under-reporting in the early days since there was no testing. The approach is based on a complex dynamic model whose parameters are estimated by an ABC algorithm (the reference being the PLoS article that initiated the ABC Wikipedia page). Results are quite interesting in that the distribution of the entry dates covers a calendar as early as December 2019 in most cases. And a proportion of missed cases as high as 99%.

“As evidence, E, we considered the cumulative number of SARS-CoV-2 cases internationally imported from China up to January 21, 2020″

The model behind remain a classical SLIR model but with a discrete and stochastic dynamical and a geographical compartmentalization based on a Voronoi tessellation centred at airports, commuting intensity and population density. Interventions by local and State authorities are also accounted for. The ABC version is a standard rejection algorithm with distance based on the evidence as quoted above. Which is a form of cdf distance (as in our Wasserstein ABC paper). For the posterior distribution of the IFR, a second ABC algorithm uses the relative distance between observed and generated deaths (per country). The paper further investigates different introduction sources (countries) before local transmission was established. For instance, China is shown to be the dominant source for the first EU countries impacted by the pandemics such as Italy, UK, Germany, France and Spain. Using a “counterfactual scenario where the surveillance systems of the US states and European countries are imagined to operate at levels able to identify 50% of all imported and locally generated infections”, the authors conclude that

*“broadening testing specifications could have considerably slowed the pandemic progression, buying considerable time to prepare mitigation responses.”*

## likelihood-free and summary-free?

Posted in Books, Mountains, pictures, Statistics, Travel with tags ABC, arXiv, Australia, Cramèr-von Mises distance, curse of dimensionality, energy, Gaussian mixture, indirect inference, information, kernel density estimator, likelihood-free methods, mean discrepancy, summary statistics, Wasserstein distance on March 30, 2021 by xi'an**M**y friends and coauthors Chris Drovandi and David Frazier have recently arXived a paper entitled *A comparison of likelihood-free methods with and without summary statistics*. In which they indeed compare these two perspectives on approximate Bayesian methods like ABC and Bayesian synthetic likelihoods.

“A criticism of summary statistic based approaches is that their choice is oftenad hocand there will generally be an inherent loss of information.”

In ABC methods, the recourse to a summary statistic is often advocated as a “necessary evil” against the greater evil of the curse of dimension, paradoxically providing a faster convergence of the ABC approximation (Fearnhead & Liu, 2018). The authors propose a somewhat generic selection of summary statistics based on [my undergrad mentors!] Gouriéroux’s and Monfort’s indirect inference, using a mixture of Gaussians as their auxiliary model. Summary-free solutions, as in our Wasserstein papers, rely on distances between distributions, hence are functional distances, that can be seen as dimension-free as well (or criticised as infinite dimensional). Chris and David consider energy distances (which sound very much like standard distances, except for averaging over all permutations), maximum mean discrepancy as in Gretton et al. (2012), Cramèr-von Mises distances, and Kullback-Leibler divergences estimated via one-nearest-neighbour formulas, for a univariate sample. I am not aware of any degree of theoretical exploration of these functional approaches towards the precise speed of convergence of the ABC approximation…

“We found that at least one of the full data approaches was competitive with or outperforms ABC with summary statistics across all examples.”

The main part of the paper, besides a survey of the existing solutions, is to compare the performances of these over a few chosen (univariate) examples, with the exact posterior as the golden standard. In the g & k model, the Pima Indian benchmark of ABC studies!, Cramèr does somewhat better. While it does much worse in an M/G/1 example (where Wasserstein does better, and similarly for a stereological extremes example of Bortot et al., 2007). An ordering inversed again for a toad movement model I had not seen before. While the usual provision applies, namely that this is a simulation study on unidimensional data and a small number of parameters, the design of the four comparison experiments is very careful, eliminating versions that are either too costly or too divergence, although this could be potentially criticised for being unrealistic (i.e., when the true posterior is unknown). The computing time is roughly the same across methods, which essentially remove the call to kernel based approximations of the likelihood. Another point of interest is that the distance methods are significantly impacted by transforms on the data, which should not be so for intrinsic distances! Demonstrating the distances are not intrinsic…