Archive for the Statistics Category

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);

common derivation for Metropolis–Hastings and other MCMC algorithms

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , on July 25, 2016 by xi'an

Khoa Tran and Robert Kohn from UNSW just arXived a paper on a comprehensive derivation of a large range of MCMC algorithms, beyond Metropolis-Hastings. The idea is to decompose the MCMC move into

  1. a random completion of the current value θ into V;
  2. a deterministic move T from (θ,V) to (ξ,W), where only ξ matters.

If this sounds like a new version of Peter Green’s completion at the core of his 1995 RJMCMC algorithm, it is bedowntown Sydney from under Sydney Harbour bridge, July 15, 2012cause it is indeed essentially the same notion. The resort to this completion allows for a standard form of the Metropolis-Hastings algorithm, which leads to the correct stationary distribution if T is self-inverse. This representation covers Metropolis-Hastings algorithms, Gibbs sampling, Metropolis-within-Gibbs and auxiliary variables methods, slice sampling, recursive proposals, directional sampling, Langevin and Hamiltonian Monte Carlo, NUTS sampling, pseudo-marginal Metropolis-Hastings algorithms, and pseudo-marginal Hamiltonian  Monte Carlo, as discussed by the authors. Given this representation of the Markov chain through a random transform, I wonder if Peter Glynn’s trick mentioned in the previous post on retrospective Monte Carlo applies in this generic setting (as it could considerably improve convergence…)

Bayesian empirical likelihood

Posted in Books, pictures, Statistics with tags , , , , , , on July 21, 2016 by xi'an

non-tibetan flags in Pula, Sardinia, June 12, 2016Sid Chib, Minchul Shin, and Anna Simoni (CREST) recently arXived a paper entitled “Bayesian Empirical Likelihood Estimation and Comparison of Moment Condition Models“. That Sid mentioned to me in Sardinia. The core notion is related to earlier Bayesian forays into empirical likelihood pseudo-models, like Lazar (2005) or our PNAS paper with Kerrie Mengersen and Pierre Pudlo. Namely to build a pseudo-likelihood using empirical likelihood principles and to derive the posterior associated with this pseudo-likelihood. Some novel aspects are the introduction of tolerance (nuisance) extra-parameters when some constraints do not hold, a maximum entropy (or exponentially tilted) representation of the empirical  likelihood function, and a Chib-Jeliazkov representation of the marginal likelihood. The authors obtain a Bernstein-von Mises theorem under correct specification. Meaning convergence. And another one under misspecification.

While the above Bernstein-von Mises theory is somewhat expected (if worth deriving) in the light of frequentist consistency results, the paper also considers a novel and exciting aspect, namely to compare models (or rather moment restrictions) by Bayes factors derived from empirical likelihoods. A grand (encompassing) model is obtained by considering all moment restrictions at once, which first sounds like more restricted, except that the extra-parameters are there to monitor constraints that actually hold. It is unclear from my cursory read of the paper whether priors on those extra-parameters can be automatically derived from a single prior. And how much they impact the value of the Bayes factor. The consistency results found in the paper do not seem to depend on the form of priors adopted for each model (for all three cases of both correctly, one correctly and none correctly specified models). Except maybe for some local asymptotic normality (LAN). Interestingly (?), the authors consider the Poisson versus Negative Binomial test we used in our testing by mixture paper. This paper is thus bringing a better view of the theoretical properties of a pseudo-Bayesian approach based on moment conditions and empirical likelihood approximations. Without a clear vision of the implementation details, from the parameterisation of the constraints (which could be tested the same way) to the construction of the prior(s) to the handling of MCMC difficulties in realistic models.

Chateau Mussar

Posted in pictures, Statistics, Travel, Wines with tags , , , , , on July 20, 2016 by xi'an

and another position in Oxford

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , on July 20, 2016 by xi'an

St. Edmund Hall, Oxford, Feb. 01, 2016After a position in Bristol advertised a few days ago, I want to point out there also is a position opening in Oxford, Department of Statistics, in conjunction with a fellowship from University College. The deadline is August 26, 2016, and applicants should contact Professor Arnaud Doucet for details.

position in Bristol

Posted in Statistics, University life with tags , , , , , , on July 19, 2016 by xi'an

Clifton Bridge, Bristol, Sept. 24, 2012There is an opening for a Lecturer (i.e., assistant/associate professor) position in Statistical Science at the University of Bristol (School of Mathematics) with deadline August 7. Please contact Professor Christophe Andrieu for more details.

computer strategies for complex Bayesian models

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , on July 18, 2016 by xi'an

frontThis is the cover page of Marco Banterle‘s thesis, who will defend on Thursday [July 21, 13:00], at a rather quiet time for French universities, which is one reason for advertising it here. The thesis is built around several of Marco’s papers, like delayed acceptance, dimension expansion, and Gaussian copula for graphical models. The defence is open to everyone, so feel free to join if near Paris-Dauphine!

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