*[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.

Although it surely could have been done analytically, I decided to tackle this problem with the brute force of simulation and, in few seconds, I got the answer that is summarized by next Table.

Rows refer to teams in the group of winners while columns are dedicated to runner-up teams. In each cell we have the probability that a match between the two teams corresponding to row and column, will be drawn. For example, the probability that Barcelona and Juventus will play a re-match of last year final is approximated by 0.133. A zero in the table indicates that the corresponding match cannot be played since it would violate one of the aforementioned constraints (e.g. Zenit and Dynamo Kyiv cannot meet due to rule 3). As an example, I report the bar graphs displaying the distribution for the next opponent of Arsenal (top), Barcelona (center) and Juventus (bottom).

The most likely match turns out to be Zenit-Arsenal with an approximated probability of 0.232: this can be explained by the fact that Zenit has only 6 and Arsenal only 5 possible opponents.

Finally, if I had to bet on the outcome of next Monday’s draw, I would pick this list of eight matches:

Real Madrid – Roma

Wolfsburg – Benfica

Atlético Madrid – Paris Saint-Germain

Manchester City – Dynamo Kyiv

Barcelona – PSV Eindhoven

Bayern München – Gent

Chelsea – Juventus

Zenit – Arsenal

According to the probabilities reported in the above Table, this is the most likely list of eight matches between all the outcomes that I observed in my simulation. Be aware though that the chances of winning the bet are very low since I observed this exact outcome only 110 times out of one million simulated draws!

Bernardo

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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

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“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

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El Capitan is a very nice mountain. It’s also the latest OS X version which messes things up with . Be aware of this before you update. I wasn’t!

I quote from a fix explained here:

Under OS X 10.11, El Capitan, writing to “/usr” is no longer allowed, even with Administrator privileges. The usual symbolic link to the active Distribution, “/usr/texbin”, is therefore removed (if it was there from a previous OS version) and cannot be installed. Many GUI applications have the path to those binaries set to “/usr/texbin” by default and will no longer find the binaries there.

I had to reinstall MacTex, then to update my GUI application (texmaker) for and finally to replace every “/usr/texbin” by “/Library/TeX/texbin”, as shown below.

Cheers

Julyan

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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.

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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while everyone was away in July, James Ridgway and I posted our “leave (the) pima paper alone” paper on arxiv, in which we discuss to which extent probit/logit regression and not too big datasets (such as the now famous Pima Indians dataset) constitute a relevant benchmark for Bayesian computation.

The actual title of the paper is “Leave Pima Indians alone…”, but xian changed it to “Leave *the* Pima Indians alone…” when discussing it on his blog. Any opinion on whether it does sound better with “the”?

On a different note, one of our findings is that Expectation-Propagation works wonderfully for such models; yes it is an approximate method, but it is very fast, and the approximation error is consistently negligible on all the datasets we looked at.

James has just posted on CRAN the EPGLM package, which computes an EP approximation of the posterior of a logit or probit model. The documentation is a bit terse at the moment, but it is very straightforward to use.

Comments on the package, the paper, its grammar or Pima Indians are most welcome!

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This very fine title quotes a pretty hilarious banquet speech by David Dunson at the last BNP conference held in Raleigh last June. The graph is by François Caron who used it in his talk there. See below for his explanation.

After the summer break, back to work. The academic year to come looks promising from a BNP point of view. Not least that three special issues have been announced, in Statistics & Computing (guest editors: Tamara Broderick (MIT), Katherine Heller (Duke), Peter Mueller (UT Austin)), the Electronic Journal of Statistics (guest editor: Subhashis Ghoshal (NCSU)), and in the International Journal of Approximate Reasoning (proposal deadline December 1st, guest editors: Alessio Benavoli (Lugano), Antonio Lijoi (Pavia) and Antonietta Mira (Lugano)).

BNP is also going to infiltrate MCMSki V, Lenzerheide, Switzerland, January 4-7 2016, with three sessions with a BNP flavor, in addition to plenary speakers David Dunson and Michael Jordan. The International Society for Bayesian Analysis World Meeting, 13 -17 June, 2016, should also host plenty of BNP sessions. And a De Finetti Lecture by Persi Diaconis (Stanford University).

Below, François’ description of his graph

- nodes are speakers at BNP9 and / or BNP10
- edges link co-authors
- node and text sizes are proportional to node degree (nb of co-authors)
- visualization with gephi (spatialization Yifan Hu)

Some comments (by François)

- it’s most probable that he missed connections
- there’s obviously a selection bias by only taking on speakers of the last two BNP meetings
- the graph is obtained by a simple “one-mode projection” of the bipartite graph authors-articles; this projection isn’t optimal since two authors of a six authors paper may not have really collaborated; Newman proposed another type of projection which weights by the number of co-authors (eg a weight of 1/3 each for a three authors publication)

Julyan

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Adam Johansen, Thomas Schön and me co-organised SMC2015, a workshop on Sequential Monte Carlo method that took place at ENSAE last week. In case you missed it, I’ve just uploaded the slides of most talks here. Enjoy!

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With colleagues Stefano Favaro and Bernardo Nipoti from Turin and Yee Whye Teh from Oxford, we have just arXived an article on discovery probabilities. If you are looking for some info on a space shuttle, a cycling team or a TV channel, it’s the wrong place. Instead, discovery probabilities are central to ecology, biology and genomics where data can be seen as a population of individuals belonging to an (ideally) infinite number of species. Given a sample of size , the -discovery probability is the probability that the next individual observed matches a species with frequency in the -sample. For instance, the probability of observing a new species is key for devising sampling experiments.

By the way, why Alan Turing? Because with his fellow researcher at Bletchley Park Irving John Good, starred in The Imitation Game too, Turing is also known for the so-called *Good-Turing estimator* of the discovery probability

which involves , the number of species with frequency in the sample (ie frequencies frequency, if you follow me). As it happens, this estimator defined in Good 1953 Biometrika paper became wildly popular among ecology-biology-genomics communities since then, at least in the small circles where wild popularity and probability aren’t mutually exclusive.

Simple explicit estimators of discovery probabilities in the Bayesian nonparametric (BNP) framework of Gibbs-type priors were given by Lijoi, Mena and Prünster in a 2007 Biometrika paper. The main difference between the two estimators of is that Good-Turing involves and only, while the BNP involves , (instead of ), and , the total number of observed species. It has been shown in the literature that the BNP estimators are more reliable than Good-Turing estimators.

How do we contribute? (i) we describe the posterior distribution of the discovery probabilities in the BNP model, which is pretty useful for deriving exact credible intervals of the estimates, and (ii) we investigate large asymptotic behavior of the estimators.

We are not aware of any non-asymptotic method for deriving credible interval for the BNP estimators. We fill this gap by describing the posterior distribution of . More specifically, we derive all posterior moments of . Since this random variable has a compact support, , it is characterized by its moments. So one can use a moment-based technique for sampling draws, see e.g. our momentify R package written for another article. We also show that the posterior distribution is explicit in two special cases of Gibbs-type priors known as the two parameter Poisson-Dirichlet prior and the normalized generalized Gamma prior. The posterior distribution is in fact shamelessly simple (once you know it) since it essentially amounts to [[a random] fraction of] a Beta distribution [with random coefficients].

As for large asymptotic behavior of the estimators, we prove the following asymptotic equivalences, denoted by ,

and for ,

where is a parameter of the Gibbs-type prior. These can serve as approximations. In the cases of the two parameter Poisson-Dirichlet prior and the normalized generalized Gamma prior, we provide also a second order term to the asymptotic expansion of the estimators

and for ,

where the second order is either a constant, or a quantity which converges almost surely to a random variable. In both cases, we show that it involves the second (and last) parameter of the priors, whereas the asymptotic equivalence given before involves only . Whether similar asymptotic expansions also hold in the whole Gibbs-type class remains an open problem!

If you have read till this point, then you may also be interested in listening to Stefano Favaro about it at the 10th Conference on Bayesian Nonparametrics next week in Raleigh, NC :-)

Cheers,

Julyan

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Hi there !

Unfortunately this post is indeed about statistics…

If you are randomly walking around the statistics blogs, you probably have certainly heard of this new language called Julia. It is said by the developers to be as easy to write as R and as fast as C (!) which is quite a catchy way of selling their work. After talking with a Julia enthusiastic user in Amsterdam, I decided to give it a try. And here I am sharing my first impressions.

Fist thing first, the installation is as easy as any other language, plus there is a neat Package management that allows you to get started quite easily. In this respect it is very similar to R.

On the minus side I became a big fan of RStudio Julian (… oupsy Julyan) told you about a long time ago. These kind of programs really make your life easier. I thus tried Juno which turned out to be cumbersome and terribly slow. I would have loved to have an IDE for Julia that would be up to the RStudio standard. Nevermind.

No lets talk a little about what is really interesting : “Is their catch phrase false advertising or not?!”.

There is a bunch of relatively good tutorials online which are really helpful to learn the basic vocabulary, but indeed if like me you are use to code in R and/or Python, you should get it pretty fast and can almost copy-paste your favourite code into Julia and with a few adjustments, it will work. So as easy to write as R : quite so.

I then tried to compare computational times for some of my latest codes and there came the good surprise ! A code that would take a handful of minutes to run in R mainly due to unavoidable loops took a couple of seconds to run in Julia, without any other sorts of optimization. The handling of big objects is smooth and I did not ran into memory problems that R was suffering from.

So far so good ! But of course there has to be some drawbacks. The first one is the poor package repository compare to CRAN or even what you can get for Python. This might of course improve in the next few years as the language is still quite new. However, it is bothering to have to re-code something when you are used to simply load a package in R. Another, probably less important problem, is the lack of data visualization methods and especially the absence of ggplot2 that we have grown quite found of around here. There is of course Gadfly, which is quite close but once again, it is up to now very limited compared to what I was used to…

All in all, I am happy to have tried Julia, and I am quite sure that I will be using it quite a lot from now on. However, even if from a efficiency point of view, it is great, and it is way easier to learn than C (which I should have done a while ago), R and its tremendous package repository is far from beaten.

Oh and by the way, it uses PyPlot based on MatplotLib that allow you to make some xkcd-like plots, which can make your presentations a lot more fun.

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