[ibayesclub.beta] reminder: Bayes club today from 2 pm (CET)

[Szabo, B.T.]

Dear  All,

This is a kind reminder that today from 2 pm (CET) we have the next meeting of the International Bayes Club (https://www.math.vu.nl/thebayesclub/). We have two speakers: Zacharie Naulet (Université Paris-Sud) and Emanuele Dolera (Pavia). The seminar will be over zoom. Please find below the titles and the abstracts, and link to the meeting.

After the talks, we will open up our virtual meeting place at gather.town so if you want to hang out and perhaps have a beer virtually with your colleagues then you are welcome to join.


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Topic: International Bayes club
Time: Apr 9, 2021 02:00 PM Amsterdam

Join Zoom Meeting
https://vu-live.zoom.us/j/93630288944?pwd=QnRMSUwwaFA1anYzWWZqdUN5WDZ5dz09

Meeting ID: 936 3028 8944
Passcode: 049612


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Link to the meeting place at gather.town:
Time: Apr 9, 2021 04:00 PM Amsterdam

https://gather.town/app/4lK8Zj28Ys0WtutP/IBC
Password: bayesbayesbayes

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To join the Bayes Club mailing list please go to:
https://listserver.vu.nl/mailman/listinfo/ibayesclub.beta

<https://listserver.vu.nl/mailman/listinfo/ibayesclub.beta>
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14:00 Speaker: Zacharie Naulet  (Université Paris-Sud)

Title: Minimax and Bayesian estimation of the unseen under tail-index regularity
Joint work with Stefano Favaro.

In this talk, I will discuss the famous problem of estimating the number of
unseen species in a population. This problem has a long history in statistics,
but has recently received a lot of attention after the breakthrough of Orlitsky
et al. who established impressive information-theoretical limits of
predictability of the unseen. For a suitable notion of loss, they proved that
minimax estimation of the unseen over all population distributions from a sample
of size $n$ is possible if and only if the population size is at most $o(n
\log(n))$. Their result is interesting in many ways, but perhaps deceptive for
statisticians, as the estimator they provide works only for large sample sizes.
This leaves open the problem of what can be done if we are willing to assume
regularity on the population distribution. On the Bayesian side, the literature
has largely investigated random partition models, giving many ways of estimating
the unseen. Most famous models, such as Poisson-Kingman partitions or Pitman-Yor processes, generate populations that possess the property of having a finite "tail-index" which entirely determine the asymptotic behaviour of the number of species. Inspired by this long line of works, we do a minimax analysis of the unseen problem over classes of population distributions having a finite
tail-index $\alpha \in (0,1)$. Albeit our analysis is mostly frequentist, the upper
bound is derived by constructing an estimator using BNP arguments and can be
easily extended to the classical Pitman-Yor estimates. Importantly, our
estimator can be efficiently computed, which I will support with some
simulations. The main challenge is on deriving the minimax lower bound. For this
matter, we propose a generic machinery for obtaining minimax lower bounds in
partition models which is of interest by itself as it can be used for many other
quantities. Interestingly, this machinery also relies on BNP arguments. In the
end, our main result is that estimating the tail-index and the unseen are
equivalent problems, and under suitable second-order assumptions on the
tail-index, minimax estimation of the unseen is possible all the way up to
population sizes that are as large as $o(\exp(c n^{\alpha})$ and impossible for larger
populations, in contrast with the $o(n \log(n))$ limit under no regularity
assumption.


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15:00 Speaker: Emanuele Dolera (Pavia)

Title: A new approach to Posterior Contraction Rates via Wasserstein dynamics

Abstract: We presents a new approach to the problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics. See [1]. Our approach relies on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs. Both the results exploit a local Lipschitz-continuity of the posterior distribution on some sufficient statistic of the data (noteworthy, the empirical distribution). See also [2]. The first contribution involves the dynamic formulation of Wasserstein distance due to Benamou and Brenier referred to as Wasserstein dynamics in order to establish PCRs under dominated Bayesian statistical models. As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both stating and proving PCRs, and it sets forth a natural connection between PCRs and three well- known problems in probability theory: the speed of mean Glivenko-Cantelli convergence, the estimation of weighted Poincare'-Wirtinger constants and Sanov large deviation principle for Wasserstein distance. The second contribution combines the use of Wasserstein distance with a suitable sieve construction to establish PCRs under full Bayesian nonparametric models. As a novelty with respect to existing literature of PCRs, our second result provides with the first treatment of PCRs under non-dominated Bayesian models. Applications of our results are presented for some classical Bayesian statistical models. By way of example, for the former result we discuss density estimation in a setting similar to that in [3], while for the latter we consider the Ferguson-Dirichlet process.

[1] Dolera, E., Favaro, S. and Mainini, E. (2020). A new approach to Posterior Contraction Rates via Wasserstein dynamics. ArXiv:2011.14425.
[2] Dolera, E. and Mainini, E. (2020). Lipschitz continuity of probability kernels in the optimal transport framework. ArXiv:2010.08380.
[3] Sriperumbudur, B., Fukumizu, K., Gretton, A., Hyvarinen, A. and Kumar, R. (2017). Density Estimation in Infinite Dimensional Exponential Families. Journal of Machine Learning Research 18, 1-59.


Best wishes,

Botond
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