It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. We start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study four concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem, a nonparametric density estimation problem and a univariate median estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.
This is joint work with Mengchu Li and Yi Yu.[-]

The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the frequentist questions of well-specification and consistency, or the numerical analysis questions of stability and well-posedness with respect to perturbations of the prior, the likelihood, or the data. This talk will outline positive and negative results (both classical ones from the literature and new ones due to the authors) on the accuracy of Bayesian inference. There will be a particular emphasis on the consequences for high- and infinite-dimensional complex systems. In particular, for such systems, subtle details of geometry and topology play a critical role in determining the accuracy or instability of Bayesian procedures. Joint with with Houman Owhadi and Clint Scovel (Caltech).[-]