**18/1, Emilia Pompe, University of Oxford: Robust inference using Posterior Bootstrap **

Abstract: Bayesian inference is known to provide misleading uncertainty estimation when the considered model is misspecified. This talk will explore various alternatives to standard Bayesian inference under model misspecification, based on extensions of the Weighted Likelihood Bootstrap (Newton & Raftery, 1994).

In the first part, we will talk about Posterior Bootstrap, which is an extension of Weighted Likelihood Bootstrap allowing the user to properly incorporate the prior. We will see how Edgeworth expansions can be used to understand the impact of the prior and guide the choice of hyperparameters.

Next we will talk about Bayesian models built of multiple components having shared parameters. Misspecification of any part of the model might propagate to all other parts and lead to unsatisfactory results. Cut distributions have been proposed as a remedy, where the information is prevented from flowing along certain directions. We will show that asymptotically cut distributions don't have the correct frequentist coverage for the associate credible regions. We will then discuss our new alternative methodology, based on the Posterior Bootstrap, which delivers credible regions with the nominal frequentist asymptotic coverage.

The talk is based on the following papers:

https://arxiv.org/abs/2103.14445

https://arxiv.org/abs/2110.11149 (joint work with Pierre Jacob)

**25/1, Tomáš Mrkvička, University of South Boemia: What can be proven by functional test statistics: The R package GET**

Abstract: Statistical testing is one of the major tools in biostatistics. Usually, the test statistic is one-dimensional, gathering the information from time or space into a single number. Performing the statistical inference with functional test statistics (such as the slope of warming measured every day of the year, spatial correlation measured in certain distances, F statistic of the GLM measured in every voxel of the brain) can reveal more information than the single agglomerative test statistic. On the other hand, using functional test statistics brings difficulties in the test statistic model assumptions, such as normality or homogeneity. Therefore, we have introduced a powerful, nonparametric statistical inference method with functional test statistics in our R package GET. The methods also provide graphical inference, which is equivalent to the formal inference, which allows for easy interpretation of the results. The package provides inference for functional GLM with one-, two- or three-dimensional functions, goodness-of-fit test based on multiple functional test statistics, graphical comparison of several distribution functions, graphical functional clustering, graphical test of dependence of two variables, functional central region detection together with functional box plot. It also allows for composite hypothesis testing in goodness of fit testing, i.e., when the model parameters must be estimated. All the procedures satisfy the family wise error rate control. We are recently working on false discovery rate control to detect all hypothesis (domain of functional test statistic) that should be rejected. The provided procedures are based on the ordering of functions according to extreme rank length functional depth, which allows for intrinsic graphical interpretation. The intrinsic graphical interpretation means that if the functional test statistic lies in at least one point outside the constructed envelopes, the null hypothesis is rejected. Thereafter, it identifies the domain of rejection.

-Myllymäki M., Mrkvička T. (2019). GET: Global envelopes in R. http://arxiv.org/abs/1911.06583

- Myllymäki M., Mrkvička T., Seijo H., Grabarnik P., Hahn U. (2017). Global envelope tests for spatial processes, JRSS Series B 79/2, 381-404.

- Mrkvička T., Roskovec T., Rost M. (2021). A nonparametric graphical tests of significance in functional GLM, Methodol Comput Appl Probab. 23, 593-612.