Archive for concentration inequalities

Markov Chains [not a book review]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , on January 14, 2019 by xi'an

As Randal Douc and Éric Moulines are both very close friends and two authors of this book on Markov chains,  I cannot engage into a regular book review! Judging from the table of contents, the coverage is not too dissimilar to the now classic Markov chain Stochastic Stability book by Sean Meyn and the late Richard Tweedie (1994), called the Bible of Markov chains by Peter Glynn, with more emphasis on convergence matters and a more mathematical perspective. The 757 pages book also includes a massive appendix on maths and probability background. As indicated in the preface, “the reason [the authors] thought it would be useful to write a new book is to survey some of the developments made during the 25 years that have elapsed since the publication of Meyn and Tweedie (1993b).” Connecting with the theoretical developments brought by MCMC methods. Like subgeometric rates of convergence to stationarity, sample paths, limit theorems, and concentration inequalities. The book also reflects on the numerous contributions of the authors to the field. Hence a perfect candidate for teaching Markov chains to mathematically well-prepared. graduate audiences. Congrats to the authors!

asymptotic properties of Approximate Bayesian Computation

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , on July 26, 2016 by xi'an

Street light near the St Kilda Road bridge, Melbourne, July 21, 2012With David Frazier and Gael Martin from Monash University, and with Judith Rousseau (Paris-Dauphine), we have now completed and arXived a paper entitled Asymptotic Properties of Approximate Bayesian Computation. This paper undertakes a fairly complete study of the large sample properties of ABC under weak regularity conditions. We produce therein sufficient conditions for posterior concentration, asymptotic normality of the ABC posterior estimate, and asymptotic normality of the ABC posterior mean. Moreover, those (theoretical) results are of significant import for practitioners of ABC as they pertain to the choice of tolerance ε used within ABC for selecting parameter draws. In particular, they [the results] contradict the conventional ABC wisdom that this tolerance should always be taken as small as the computing budget allows.

Now, this paper bears some similarities with our earlier paper on the consistency of ABC, written with David and Gael. As it happens, the paper was rejected after submission and I then discussed it in an internal seminar in Paris-Dauphine, with Judith taking part in the discussion and quickly suggesting some alternative approach that is now central to the current paper. The previous version analysed Bayesian consistency of ABC under specific uniformity conditions on the summary statistics used within ABC. But conditions for consistency are now much weaker conditions than earlier, thanks to Judith’s input!

There are also similarities with Li and Fearnhead (2015). Previously discussed here. However, while similar in spirit, the results contained in the two papers strongly differ on several fronts:

  1. Li and Fearnhead (2015) considers an ABC algorithm based on kernel smoothing, whereas our interest is the original ABC accept-reject and its many derivatives
  2. our theoretical approach permits a complete study of the asymptotic properties of ABC, posterior concentration, asymptotic normality of ABC posteriors, and asymptotic normality of the ABC posterior mean, whereas Li and Fearnhead (2015) is only concerned with asymptotic normality of the ABC posterior mean estimator (and various related point estimators);
  3. the results of Li and Fearnhead (2015) are derived under very strict uniformity and continuity/differentiability conditions, which bear a strong resemblance to those conditions in Yuan and Clark (2004) and Creel et al. (2015), while the result herein do not rely on such conditions and only assume very weak regularity conditions on the summaries statistics themselves; this difference allows us to characterise the behaviour of ABC in situations not covered by the approach taken in Li and Fearnhead (2015);

métro static

Posted in Kids, Statistics, Travel with tags , , , , , , on July 19, 2015 by xi'an

[In the train shuttle at Birmingham airport, two young guys, maybe back from SPA 2015, discussing signal processing:]

– In Bayesian statistics, they use a different approach to testing hypotheses… You see, they put priors on the different hypotheses…

– But in the end it all boils down to concentration inequalities…