## 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 Gibbs-type 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 Gibbs-type 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 Gibbs-type priors which encompasses the two parameter Poisson-Dirichlet 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 0-mean 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 rate-optimal 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. Schmidt-Hieber [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…)

## Back to blogging

My last post dates back to May 2015… thanks to JB and Julyan for keeping the place busy! I’m not (quite) dead and intend to go back to posting stuff every now and then. And by the way, congrats to both for their new jobs!

Last July, I’ve also started a new job, as an assistant professor in the Department of Statistics at Harvard University, after having spent two years in Oxford. At some point, I might post something on the cultural difference between the ~~European~~ English and American communities of statisticians.

In the coming weeks, I’ll tell you all about a new paper entitled Coupling of Particle Filters, co-written with Fredrik Lindsten and Thomas B. Schön from Uppsala University in Sweden. We are excited about this coupling idea because it’s simple and yet brings massive gains in many important aspects of inference for state space models (including both parameter inference and smoothing). I’ll be talking about it at the World Congress in Probability and Statistics in Toronto next week and at JSM in Chicago, early in August.

I’ll also try to write about another exciting project, joint work with Christian Robert, Chris Holmes and Lawrence Murray, on modularization, cutting feedback, the infamous cut function of BUGS and all that funny stuff. I’ve talked about it in ISBA 2016, and intend to put the associated tech report on arXiv over the summer.

Stay tuned!

## 3D density plot in R with Plotly

In Bayesian nonparametrics, many models address the problem of *density regression*, including covariate dependent processes. These were settled by the pioneering works by [current ISBA president] MacEachern (1999) who introduced the general class of dependent Dirichlet processes. The literature on dependent processes was developed in numerous models, such as nonparametric regression, time series data, meta-analysis, to cite but a few, and applied to a wealth of fields such as, e.g., epidemiology, bioassay problems, genomics, finance. For references, see for instance the chapter by David Dunson in the Bayesian nonparametrics textbook (edited in 2010 by Nils Lid Hjort, Chris Holmes, Peter Müller and Stephen G. Walker). With Kerrie Mengersen and Judith Rousseau, we have proposed a dependent model in the same vein for modeling the influence of fuel spills on species diversity (arxiv).

Several densities can be plotted on the same 3D plot thanks to the Plotly R library, *“an interactive, browser-based charting library built on the open source JavaScript graphing library, plotly.js.”*

In our ecological example, the model provides a series of densities on the *Y* axis (in our case, posterior density of species diversity), indexed by some covariate *X* (a pollutant). See file density_plot.txt. The following Plotly R code

library(plotly) mydata = read.csv("density_plot.txt") df = as.data.frame(mydata) plot_ly(df, x = Y, y = X, z = Z, group = X, type = "scatter3d", mode = "lines")

provides a graph as below. For the interactive version, see the RPubs page here.

## Bayesian demography

“For about two centuries, Bayesian demography remained largely dormant. Only in recent decades has there been a revival of demographers’ interest in Bayesian methods, following the methodological and computational developments of Bayesian statistics. The area is currently growing fast, especially with the United Nations (UN) population projections becoming probabilistic—and Bayesian.”Bijak and Bryant (2016)

It is interesting to see that Bayesian statistics have been infiltrating demography in the recent years. The review paper Bayesian demography 250 years after Bayes by Bijak and Bryant (Population Studies, 2016) stresses that promising areas of application include *demographic forecasts, problems with limited data, and highly structured and complex models*. As an indication of this growing interest, ISBA meeting to be held next June will showcase a course and a session devoted to the field (given and organized by Adrian Raftery).

With Vianney Costemalle from INSEE, we recently modestly contributed to the field by proposing a Bayesian model (paper in French) which helps reconciling apparently inconsistent population datasets. The aim is to estimate annual migration flows to France (note that the work covers the period 2004-2011 (long publication process) and as a consequence does not take into account recent migration events). We follow the United Nations (UN) definition of a *long-term migrant, who is someone who settles in a foreign country for at least one year. *At least two datasets can be used to this aim: 1) the population census , annual since 2004, and 2) data from residence permits . (more…)

## Workshop on Bayesian Nonparametrics in Turin

On February 19 took place at Collegio Carlo Alberto the second Statalks, a series of Italian workshops aimed at Master students, PhD students, post-docs and young researchers. This edition was dedicated to Bayesian Nonparametrics. The first two presentations were introductory tutorials while the last four focused on theory and applications. All six were clearly biased according to the scientific interests of our group. Below are the program and the slides.

- A gentle introduction to Bayesian Nonparametrics I (Antonio Canale)
- A gentle introduction to Bayesian Nonparametrics II (Julyan Arbel)
- Dependent processes in Bayesian Nonparametrics (Matteo Ruggiero)
- Asymptotics for discrete random measures (Pierpaolo De Blasi)
- Applications to Ecology and Marketing (Antonio Canale)
- Species sampling models (Julyan Arbel)

## Champions League eight of finals’ draw: what are the odds?

*[This is a guest post by my friend and colleague Bernardo Nipoti from Collegio Carlo Alberto, Juventus Turin.]*

The matches of the group stage of the UEFA Champions league have just finished and next Monday, the 14th of December 2015, in Nyon, there will be a round of draws for deciding the eight matches that will compose the first round of the knockout phase.

As explained on the UEFA website, rules are simple:

- two seeding pots have been formed: one consisting of group winners and the other of runners-up;
- no team can play a club from their group or any side from their own association;
- due to a decision by the UEFA Executive Committee, teams from Russia and Ukraine cannot meet.

The two pots are:

Group winners: Real Madrid (ESP), Wolfsburg (GER), Atlético Madrid (ESP), Manchester City (ENG), Barcelona (ESP, holders), Bayern München (GER), Chelsea (ENG), Zenit (RUS);

Group runners-up: Paris Saint-Germain (FRA), PSV Eindhoven (NED), Benfica (POR), Juventus (ITA), Roma (ITA), Arsenal (ENG), Dynamo Kyiv (UKR), Gent (BEL).

Giving these few constraints, are there some matches that are more likely to be drawn than others? For example, supporters of Barcelona might wonder whether the seven possible teams (PSG, PSV, Benfica, Juventus, Arsenal, Dynamo Kyiv and Gent) are all equally likely to be the next opponent of their favorite team. (more…)

## List of predatory publishers

I have been recently invited to referee a paper for a journal I had never heard of before: the International Journal of Biological Instrumentation, published by VIBGYOR Online Publishers. This publisher happens to be on the blacklist of *predatory publishers* by Jeffrey Beall which inventory:

## Potential, possible, or probable predatory scholarly open-access publishers.

I have kindly declined the invitation. Thanks Igor for the link.

Julyan

## Some thoughts on the life of a mathematician, by Villani

Some time ago, Cédric Villani came to Turin for delivering two talks. One intended for youngsters (high school level say), another one for a wider audience, as a recipient of the Peano Prize. He commented on live, in Italian *per favore*:

“Grazie mille! Un grande piacere e un grande onore per me!”

I attended both. The reason why I attended the first being that I am acting as a research advisor for Math en Jeans groups. Villani spoke about his book, *Birth of a Theorem*, or *Théorème Vivant*. He also shared a list of se7en thoughts/tips about doing research, with illustrations. I find them quite inspiring, here they are.

**Documentation/literature**

Illustrating this by showing Faà di Bruno’s formula Wikipedia page. I like this quote, since the formula enters moment computation for objects I’m using everyday. And also because Faà di Bruno lived in Italian Piedmont, precisely in Turin.**Motivation**

*“The most important and the most mysterious.”***Favorable environment**

Showing pictures of several places where he worked, including Institut Henri Poincaré. Not sure that this one is the most favorable environment for scientific productivity (as a Director I mean).**Exchanges**

Meaning between scientists, not trade. Explaining briefly about polymath projects. And displaying a snapshot of Gowers’s Weblog as an illustration of how diverse exchanges he means. I also believe that blogs are a great information medium🙂**Constraints**

With snapshots of Musica Ricercata sheet music. And a paragraph of*La disparition*, a novel without the letter*e*by Georges Perec. Writing this makes me realize how foolish such an enterprise would look like in mathematics.**Work & Intuition**

Interesting to see these two at the same level.**Perseverance & Luck**

Same comment as for point 6.

Julyan

## Why shrinking priors shrink ?

Hello there !

While I was in Amsterdam, I took the opportunity to go and work with the Leiden crowd, an more particularly with Stéphanie van der Pas and Johannes Schmidt-Heiber. Since Stéphanie had already obtained neat results for the Horseshoe prior and Johannes had obtained some super cool results for the spike and slab prior, they were the fist choice to team up with to work on sparse models. And guess what ? we have just ArXived a paper in which we study the sparse Gaussian sequence

where only a small number of are non zero.

There is a rapidly growing literature on shrinking priors for such models, just look at Polson and Scott (2012), Caron and Doucet (2008), Carvalho, Polson, and Scott (2010) among many, many others, or simply have a look at the program of the last BNP conference. There is also an on growing literature on theoretical properties of some of these priors. The Horseshoe prior was studied in Pas, Kleijn, and Vaart (2014), an extention of the Horseshoe was then study in Ghosh and Chakrabarti (2015), and recently, the spike and slab Lasso was studied in Rocková (2015) (see also Xian ’Og)

All these results are super nice, but still we want to know **why do some shinking priors shrink so well and others do not?!** As we are *all* mathematicians here, I will reformulate this last question: **What would be the conditions on the prior under which the posterior contracts at the minimax rate ^{1}** ?

We considered a Gaussian scale mixture prior on the sequence

since this family of priors encomparse all the ones studied in the papers mentioned above (and more), so it seemed to be general enough.

Our main contribution is to give conditions on such that the posterior converge at the good rate. We showed that in order to recover the parameter that are non-zeros, the prior should have tails that decays at most exponentially fast, which is similar to the condition impose for the Spike and Slab prior. Another expected condition is that the prior should put enough mass around 0, since our assumption is that the vector of parameter is nearly black i.e. most of its components are 0.

More surprisingly, in order to recover 0 parameters correctly, one also need some conditions on the tail of the prior. More specifically, the prior’s tails cannot be too big, and if they are, we can then construct a prior that puts enough mass near 0 but which does not concentrate at the minimax rate.

We showed that these conditions are satisfied for many priors including the Horseshoe, the Horseshoe+, the Normal-Gamma and the Spike and Slab Lasso.

The Gaussian scale mixture are also quite simple to use in practice. As explained in Caron and Doucet (2008) a *simple* Gibbs sampler can be implemented to sample from the posterior. We conducted simulation study to evaluate the *sharpness* of our conditions. We computed the loss for the Laplace prior, the global-local scale mixture of gaussian (called hereafter *bad* prior for simplicity), the Horseshoe and the Normal-Gamma prior. The first two do not satisfy our condition, and the last two do. The results are reported in the following picture.

As we can see, priors that do and do not satisfy our condition show different behaviour (it seems that the priors that do not fit our conditions have a risk larger than the minimax rate of a factor of ). This seems to indicate that our conditions are sharp.

At the end of the day, our results expands the class of shrinkage priors with theoretical guarantees for the posterior contraction rate. Not only can it be used to obtain the optimal posterior contraction rate for the horseshoe+, the inverse-Gaussian and normal-gamma priors, but the conditions provide some characterization of properties of sparsity priors that lead to desirable behaviour. Essentially, the tails of the prior on the local variance should be at least as heavy as Laplace, but not too heavy, and there needs to be a sizable amount of mass around zero compared to the amount of mass in the tails, in particular when the underlying mean vector grows to be more sparse.

## Reference

Caron, François, and Arnaud Doucet. 2008. “Sparse Bayesian Nonparametric Regression.” In *Proceedings of the 25th International Conference on Machine Learning*, 88–95. ICML ’08. New York, NY, USA: ACM.

Carvalho, Carlos M., Nicholas G. Polson, and James G. Scott. 2010. “The Horseshoe Estimator for Sparse Signals.” *Biometrika* 97 (2): 465–80.

Ghosh, Prasenjit, and Arijit Chakrabarti. 2015. “Posterior Concentration Properties of a General Class of Shrinkage Estimators Around Nearly Black Vectors.”

Pas, S.L. van der, B.J.K. Kleijn, and A.W. van der Vaart. 2014. “The Horseshoe Estimator: Posterior Concentration Around Nearly Black Vectors.” *Electron. J. Stat.* 8: 2585–2618.

Polson, Nicholas G., and James G. Scott. 2012. “Good, Great or Lucky? Screening for Firms with Sustained Superior Performance Using Heavy-Tailed Priors.” *Ann. Appl. Stat.* 6 (1): 161–85.

Rocková, Veronika. 2015. “Bayesian Estimation of Sparse Signals with a Continuous Spike-and-Slab Prior.”

- For those wondering why the heck with minimax rate here, just remember that a posterior that contracts at the minimax rate induces an estimator which converge at the same rate. It also gives us that confidence region will not be too large.↩

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