Bayesian Asymptotic Theory for Manifold Data and Bayesian Smeariness of Location Statistics

• Applicant: Privatdozent Dr. Benjamin Eltzner
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 524285496

In statistical analysis on non-Euclidean data spaces and descriptor spaces, it was recently found that the classical Central Limit Theorem (CLT) is not always applicable for the Fréchet mean. This lead to generalized asymptotic theory including lower rates of convergence, called smeariness. This project aims at investigating smeariness and the related concept of finite sample smeariness in a Bayesian setting, which has not been attempted before. This leads to two major complications. Firstly, the choice of prior has a significant impact on finite sample behavior as indicated by a preliminary study shown in Figure 3, which affects finite sample smeariness. Secondly, all known examples of smeariness of the Fréchet mean occur in multimodal distributions, whereas the Fréchet mean is most naturally modeled as a parameter describing the center of a single, symmetric mode. Analogously, in the Bayesian inference on a toy model presented here, the true distribution of the data is bimodal while the family of distributions whose likelihood is used for inference is unimodal, which amounts to a model misspecification. In consequence, the Bernstein-von-Mises Theorem (BvMT) cannot be applied and therefore the posterior variance typically differs from the variance of the posterior mean. This project aims at coping with such model misspecifications. As a theoretical foundation, we aim for a generalization of the BvMT by investigating concepts like a Bayesian learning rate as introduced in the Safe Bayesian approach by Grünwald and van Ommen (2017). This serves to develop a Bayesian estimation procedure which restores the relationship between the variance of the mean and the posterior variance in case of misspecification leading to smeariness. Additionally, the influence of the prior in a Bayesian estimation setting on finite sample behavior such as finite sample smeariness will be systematically characterized. Since smeariness undermines the estimation of estimator variance and the approximation of quantiles for hypothesis tests, it is desirable to introduce location statistics with a similar interpretation to the Fréchet mean which are widely applicable and are generally non-smeary. To this end we generalize the recently described diffusion means to the Bayesian estimation setting and aim at reproducing the non-smeariness result which was achieved in the frequentist setting. While nonparametric Bayesian inference has been successfully used to avoid model misspecification, that approach implies a much wider scope than the simple location statistic setting on which we focus in this project.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr. Stephan Huckemann