ISBA elections, let’s go voting
The International Society for Bayesian Analysis (ISBA), is running elections until November, 15. This year, two contributors on this blog, Nicolas Chopin and myself, are running for an ISBA Section office. The sections of the society, nine in number as of today, gather researchers with common research interests: Computation, Objective Bayes, Nonparametrics, etc.
Here are our candidate statements:
Statistical learning in models made of modules
Hi,
With Lawrence Murray, Chris Holmes and Christian Robert, we have recently arXived a paper entitled “Better together? Statistical learning in models made of modules”. Christian blogged about it already. The context is the following: parameters of a first model appear as inputs in another model. The question is whether to consider a “joint model approach”, where all parameters are estimated simultaneously with all of the data. Or if one should instead follow a “modular approach”, where the first parameters are estimated with the first model only, ignoring the second model. Examples of modular approaches include the “cut distribution“, or “twostep estimators” (e.g. Chapter 6 of Newey & McFadden (1994)). In many fields, modular approaches are preferred, because the second model is suspected of being more misspecified than the first one. Misspecification of the second model can “contaminate” the joint model, with dire consequences on inference, as described e.g. in Bayarri, Berger & Liu (2009). Other reasons include computational constraints and the lack of simultaneous availability of all models and associated data. In the paper, we try to make sense of the defects of the joint model approach and we propose a principled, quantitative way of choosing between joint and modular approaches.
School of Statistics for Astrophysics, Autrans, France, October 913
Didier FraixBurnet (IPAG), Stéphane Girard (Inria) and myself are organising a School of Statistics for Astrophysics, Stat4Astro, to be held in October in France. The primary goal of the School is to train astronomers to the use of modern statistical techniques. It also aims at bridging the gap between the two communities by emphasising on the practice during works in common, to give firm grounds to the theoretical lessons, and to initiate works on problems brought by the participants. There have been two previous sessions of this school, one on regression and one on clustering. The speakers of this edition, including Christian Robert, Roberto Trotta and David van Dyk, will focus on the Bayesian methodology, with the moral support of the Bayesian Society, ISBA. The interest of this statistical approach in astrophysics probably comes from its necessity and its success in determining the cosmological parameters from observations, especially from the cosmic background fluctuations. The cosmological community has thus been very active in this field (see for instance the Cosmostatistics Initiative COIN).
But the Bayesian methodology, complementary to the more classical frequentist one, has many applications in physics in general due to its faculty to incorporate a priori knowledge into the inference computation, such as the uncertainties brought by the observational processes.
As for sophisticated statistical techniques, astronomers are not familiar with Bayesian methodology in general, while it is becoming more and more widespread and useful in the literature. This school will form the participants to both a strong theoretical background and a solid practice of Bayesian inference:
 Introduction to R and Bayesian Statistics (Didier FraixBurnet, Institut de Planétologie et d’Astrophysique de Grenoble)
 Foundations of Bayesian Inference (David van Dyk, Imperial College London)
 Markov chain Monte Carlo (David van Dyk, Imperial College London)
 Model Building (David van Dyk, Imperial College London)
 Nested Sampling, Model Selection, and Bayesian Hierarchical Models (Roberto Trotta, Imperial College London)
 Approximate Bayesian Computation (Christian Robert, Univ. ParisDauphine, Univ. Warwick and Xi’an (!))
 Bayesian Nonparametric Approaches to Clustering (Julyan Arbel, Université Grenoble Alpes and Inria)
Feel free to register, we are not fully booked yet!
Julyan
Particle methods in Statistics
Hi there,
In this post, just in time for the summer, I propose a reading list for people interested in discovering the fascinating world of particle methods, aka sequential Monte Carlo methods, and their use in statistics. I also take the opportunity to advertise the SMC workshop in Uppsala (30 Aug – 1 Sept), which features an amazing list of speakers, including my postdoctoral collaborator Jeremy Heng:
SubGaussian property for the Beta distribution (part 1)
With my friend Olivier Marchal (mathematician, not filmmaker, nor the cop), we have just arXived a note on the subGaussianity of the Beta and Dirichlet distributions.
The notion, introduced by JeanPierre Kahane, is as follows:
A random variable with finite mean is subGaussian if there is a positive number such that:
Such a constant is called a proxy variance, and we say that is subGaussian. If is subGaussian, one is usually interested in the optimal proxy variance:
Note that the variance always gives a lower bound on the optimal proxy variance: . In particular, when , is said to be strictly subGaussian.
The subGaussian property is closely related to the tails of the distribution. Intuitively, being subGaussian amounts to having tails lighter than a Gaussian. This is actually a characterization of the property. Let . Then:
That equivalence clearly implies exponential upper bounds for the tails of the distribution since a Gaussian satisfies
That can also be seen directly: for a subGaussian variable ,
The polynomial function is minimized on at , for which we obtain
.
In that sense, the subGaussian property of any compactly supported random variable comes for free since in that case the tails are obviously lighter than those of a Gaussian. A simple general proxy variance is given by Hoeffding’s lemma. Let be supported on with . Then for any ,
so is subGaussian.
Back to the Beta where , this shows the Beta is subGaussian. The question of finding the optimal proxy variance is a more challenging issue. In addition to characterizing the optimal proxy variance of the Beta distribution in the note, we provide the simple upper bound . It matches with Hoeffding’s bound for the extremal case , , where the Beta random variable concentrates on the twopoint set (and when Hoeffding’s bound is tight).
In getting the bound , we prove a recent conjecture made by Sam Elder in the context of Bayesian adaptive data analysis. I’ll say more about getting the optimal proxy variance in a next post soon.
Cheers!
Julyan
ABC in Banff
Hi all,
Last week I attended a wonderful meeting on Approximate Bayesian Computation in Banff, which gathered a nice crowd of ABC users and enthusiasts, including lots of people outside of computational stats, whom I wouldn’t have met otherwise. Christian blogged about it there. My talk on Inference with Wasserstein distances is available as a video here (joint work with Espen Bernton, Mathieu Gerber and Christian Robert, the paper is here). In this post, I’ll summarize a few (personal) points and questions on ABC methods, after recalling the basics of ABC (ahem).
Gaussian variates truncated to a finite interval
Alan Rogers, an anthropologist at University of Utah, got in touch with me about my paper on the simulation of truncated Gaussian distributions (journal version, arxiv version). The method I proposed in this paper works for either a finite interval [a,b], or a semifinite one [a,+inf[, but my C code implements only the latter, and Alan needed the former.
Alan thus decided to reimplement my method and several others (including Christian Robert’s acceptreject algorithm proposed in this paper) in C; see here:
https://github.com/alanrogers/dtnorm
Alan also sent me this interesting plot that compares the different methods. The color of a dot at position (a,b) corresponds to the fastest method for simulating N(0,1) truncated to [a,b];
A few personal remarks:
 My method is an acceptreject algorithm, where the proposal is a mixture of uniform distributions on rectangles. The point is to have a large probability that the number of basic operations (multiplication, division) needed to return the draw is small. However, the improvement brought by such a method might be be observable only in compiled languages. In an interpreted language such as R, Matlab and Python, loops over basic operations come with a certain overhead, which might cancel any improvement. This was the experience of a colleague who tried to implement it in Julia.
 Even in C, this comparison might depend on several factors (computer, compiler, libraries, and so on). If I remember correctly, the choice of the random generator in particular may have a significant impact. (I used the GSL library which makes it easy to try different generators for the same piece of code.)
 Also bear in mind that some progress has been made for computing the inverse CDF of a unit Gaussian distribution. Hence the basic inverse CDF method, while not being the fastest approach, works reasonably well these days, especially (again) in interpreted languages. (Update: Alan tells me the inverse CDF methods remains 10 times slower for his C implementation, based on the GSL library.)
Faà di Bruno’s note on eponymous formula, trilingual version
The Italian mathematician Francesco Faà di Bruno was born in Alessandria (Piedmont, Italy) in 1825 and died in Turin in 1888. At the time of his birth, Piedmont used to be part of the Kingdom of Sardinia, led by the Dukes of Savoy. Italy was then unified in 1861, and the Kingdom of Sardinia became the Kingdom of Italy, of which Turin was declared the first capital. At that time, Piedmontese used to commonly speak both Italian and French.
Faà di Bruno is probably best known today for the eponymous formula which generalizes the derivative of a composition of two functions, , to any order:
over tuples satisfying
Faà di Bruno published his formula in two notes:
 Faà Di Bruno, F. (1855). Sullo sviluppo delle funzioni. Annali di Scienze Matematiche e Fisiche, 6:479–480. Google Books link.

Faà Di Bruno, F. (1857). Note sur une nouvelle formule de calcul différentiel. Quarterly Journal of Pure and Applied Mathematics, 1:359–360. Google Books link.
They both date from December 1855, and were signed in Paris. They are similar and essentially state the formula without a proof. I have arXived a note which contains a translation from the French version to English (reproduced below), as well as the two original notes in French and in Italian. I’ve used for this the Erasmus MMXVI font, thanks Xian for sharing! (more…)
postdoc positions at ENSAE
Hi,
interested in doing a postdoc with me on anything related to Bayesian Computation? Please let me know, as there is currently a call for postdoc grants at the ENSAE, see below.
Nicolas Chopin
The Labex ECODEC is a research consortium in Economics and Decision Sciences common to three leading French higher education institutions based in the larger Paris area: École polytechnique, ENSAE and HEC Paris. The Labex Ecodec offers:
 Oneyear postdoctoral fellowships for 20172018

Twoyear postdoctoral fellowships for 20172019
The monthly gross salary of postdoctoral fellowships is 3 000 €.
Candidates are invited to contact as soon as possible members of the research group (see below) with whom they intend to work.
Research groups concerned by the call:
Area 1: Secure Careers in a Global Economy
Area 2: Financial Market Failures and Regulation
Area 3: Product Market Regulation and Consumer DecisionMaking
Area 4: Evaluating the Impact of Public Policies and Firms’ Decisions
Area 5: New Challenges for New Data
Details of axis can be found on the website:
Deadlines for application:
31^{st} December 2016
Screening of applications and decisions can be made earlier for srong candidates who need an early decision.
The application should be sent to application@labexecodec.fr in PDF. Please mention the area number on which you apply in the subject.
The application package includes:
 A cover letter with the name of a potential supervisor among the group;
 A research statement;
 A letter from the potential supervisor in support of the project;
 A Curriculum vita (with the address of the candidate, phone and email contact).
 The Ph.D. dissertation or papers/preprint;
 Reference letters, including one from the PhD advisor. A letter from a member of the research group with whom the candidate is willing to interact will be appreciated.
Please note that HEC, Genes, and X PhD students are not eligible to apply for this call.
Selection will be based on excellence and a research project matching the group’s research agenda.
Area 1 “Secure careers in a Global Economy”: Pierre Cahuc (ENSAE), Dominique Rouziès (HEC), Isabelle Méjean (École polytechnique)
Area 2: “Financial Market Failures and Regulation”: François Derrien (HEC), JeanDavid Fermanian, (ENSAE) Edouard Challe (École polytechnique)
Area 3: “DecisionMaking and Market Regulation”: Nicolas Vieille (HEC), Philippe Choné (ENSAE), MarieLaure Allain (École polytechnique)
Area 4: “Evaluating the Impact of Public Policies and Firm’s Decisions”: Bruno Crépon (ENSAE), Yukio Koriyama (École polytechnique), Daniel Halbheer (HEC)
Area 5: “New Challenges for New Data”: Anna Simoni (ENSAE), Gilles Stoltz (HEC)
MathSciNet reviews on Bayesian papers
I recently started to review papers on Mathematical Reviews / MathSciNet a decided I would post the reviews here from time to time. Here are the first three which deal with (i) objective Bayes priors for discrete parameters, (ii) random probability measures and inference on species variety and (iii) Bayesian nonparametric asymptotic theory and contraction rates.
 An objective approach to prior mass functions for discrete parameter spaces, by Villa, C. and Walker, S. G., J. Amer. Statist. Assoc. 110 (2015), no. 511, 1072–1082.
The paper deals with objective prior derivation in the discrete parameter setting. Previous treatment of this problem includes J. O. Berger, J.M. Bernardo and D. Sun [J. Amer. Statist. Assoc. 107 (2012), no. 498, 636–648; MR2980073] who rely on embedding the discrete parameter into a continuous parameter space and then applying reference methodology (J.M. Bernardo [J. Roy. Statist. Soc. Ser. B 41 (1979), no. 2, 113–147; MR0547240]). The main contribution here is to propose an all purpose objective prior based on the Kullback–Leibler (KL) divergence. More specifically, the prior at any parameter value is obtained as follows: (i) compute the minimum KL divergence over between models indexed by and ; (ii) set proportional to a sound transform of the minimum obtained in (i). A good property of the proposed approach is that it is not problem specific. This objective prior is derived in five models (including binomial and hypergeometric) and is compared to the priors known in the literature. The discussion suggests possible extension to the continuous parameter setting.
 A note on nonparametric inference for species variety with Gibbstype priors, by Favaro, Stefano and James, Lancelot F., Electron. J. Stat. 9 (2015), no. 2, 2884–2902.
A. Lijoi, R. H. Mena and I. Prünster [Biometrika 94 (2007), no. 4, 769–786; MR2416792] recently introduced a Bayesian nonparametric methodology for estimating the species variety featured by an additional unobserved sample of size given an initial observed sample. This methodology was further investigated by S. Favaro, Lijoi and Prünster [Biometrics 68 (2012), no. 4, 1188–1196; MR3040025; Ann. Appl. Probab. 23 (2013), no. 5, 1721–1754; MR3114915]. Although it led to explicit posterior distributions under the general framework of Gibbstype priors [A. V. Gnedin and J. W. Pitman (2005), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 12, 83–102, 244–245;MR2160320], there are situations of practical interest where is required to be very large and the computational burden for evaluating these posterior distributions makes impossible their concrete implementation. This paper presents a solution to this problem for a large class of Gibbstype priors which encompasses the two parameter PoissonDirichlet prior and, among others, the normalized generalized Gamma prior. The solution relies on the study of the large asymptotic behaviour of the posterior distribution of the number of new species in the additional sample. In particular a simple characterization of the limiting posterior distribution is introduced in terms of a scale mixture with respect to a suitable latent random variable; this characterization, combined with the adaptive rejection sampling, leads to derive a large approximation of any feature of interest from the exact posterior distribution. The results are implemented through a simulation study and the analysis of a dataset in linguistics.
 Rate exact Bayesian adaptation with modified block priors, by Gao, Chao and Zhou, Harrison H., Ann. Statist. 44 (2016), no. 1, 318–345.
A novel prior distribution is proposed for adaptive Bayesian estimation, meaning that the associated posterior distribution contracts to the truth with the exact optimal rate and at the same time is adaptive regardless of the unknown smoothness. The prior is termed \textit{block prior} and is defined on the Fourier coefficients of a curve by independently assigning 0mean Gaussian distributions on blocks of coefficients indexed by some , with covariance matrix proportional to the identity matrix; the proportional coefficient is itself assigned a prior distribution . Under conditions on , it is shown that (i) the prior puts sufficient prior mass near the true signal and (ii) automatically concentrates on its effective dimension. The main result of the paper is a rateoptimal posterior contraction theorem obtained in a general framework for a modified version of a block prior. Compared to the closely related block spike and slab prior proposed by M. Hoffmann, J. Rousseau and J. SchmidtHieber [Ann. Statist. 43 (2015), no. 5, 2259–2295; MR3396985] which only holds for the white noise model, the present result can be applied in a wide range of models. This is illustrated through applications to five mainstream models: density estimation, white noise model, Gaussian sequence model, Gaussian regression and spectral density estimation. The results hold under Sobolev smoothness and their extension to more flexible Besov smoothness is discussed. The paper also provides a discussion on the absence of an extra log term in the posterior contraction rates (thus achieving the exact minimax rate) with a comparison to other priors commonly used in the literature. These include rescaled Gaussian processes [A. W. van der Vaart and H. van Zanten, Electron. J. Stat. 1 (2007), 433–448; MR2357712; Ann. Statist. 37 (2009), no. 5B, 2655–2675; MR2541442] and sieve priors [V. Rivoirard and J. Rousseau, Bayesian Anal. 7 (2012), no. 2, 311–333; MR2934953; J. Arbel, G. Gayraud and J. Rousseau, Scand. J. Stat. 40 (2013), no. 3, 549–570; MR3091697].
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