{"success":1,"msg":"","color":"rgb(28, 35, 49)","title":"A new approach to Posterior Contraction Rates via Wasserstein dynamics<\/b>","description":"webinar","title2":"","start":"2021-04-09 15:00","end":"2021-04-09 16:00","responsable":"Botond Szabo <\/i><\/a>","speaker":"Emanuele Dolera (Pavia)","id":"35","type":"webinar","timezone":"Europe\/Amsterdam","activity":"Zoom (Meeting ID: 916 2981 6385, Passcode: 049612)","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."}