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).
Statistical inference with the Wasserstein distance
Hi! It’s been too long!
In a recent arXiv entry, Espen Bernton, Mathieu Gerber and Christian P. Robert and I explore the use of the Wasserstein distance to perform parameter inference in generative models. A byproduct is an ABCtype approach that bypasses the choice of summary statistics. Instead, one chooses a metric on the observation space. Our work fits in the minimum distance estimation framework and is particularly related to “On minimum Kantorovich distance estimators”, by Bassetti, Bodini and Regazzini. A recent and very related paper is “Wasserstein training of restricted Boltzmann machines“, by Montavon, Müller and Cuturi, who have similar objectives but are not considering purely generative models. Similarly to that paper, we make heavy use of recent breakthroughs in numerical methods to approximate Wasserstein distances, breakthroughs which were not available to Bassetti, Bodini and Regazzini in 2006.
Here I’ll describe the main ideas in a simple setting. If you’re excited about ABC, asymptotic properties of minimum Wasserstein estimators, Hilbert spacefilling curves, delay reconstructions and Takens’ theorem, or SMC samplers with rhit kernels, check our paper!
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].
Collegio Carlo Alberto
I have spent three years as a postdoc at the Collegio Carlo Alberto. This was a great time during which I have been able to interact with top colleagues and to prepare my applications in optimal conditions. Now that I have left for Inria Grenoble, here is a brief picture presentation of the Collegio. (more…)
Coupling of particle filters: smoothing
Hi again!
In this post, I’ll explain the new smoother introduced in our paper Coupling of Particle Filters with Fredrik Lindsten and Thomas B. Schön from Uppsala University. Smoothing refers to the task of estimating a latent process of length , given noisy measurements of it, ; the smoothing distribution refers to . The setting is statespace models (what else?!), with a fixed parameter assumed to have been previously estimated.
Coupling of particle filters: likelihood curves
Hi!
In this post, I’ll write about coupling particle filters, as proposed in our recent paper with Fredrik Lindsten and Thomas B. Schön from Uppsala University, available on arXiv; and also in this paper by colleagues at NUS. The paper is about a methodology with multiple direct consequences. In this first post, I’ll focus on correlated likelihood estimators; in a later post, I’ll describe a new smoothing algorithm. Both are described in detail in the article. We’ve been blessed to have been advertised by xi’an’s og, so glory is just around the corner.
leave a comment