Sofia Olhede - On Graph Limits as Models for Interaction Data
Network data has become a staple in many different applications, ranging from ecology, to neuroscience and systems biology. Its inference will of course depend on the application where we collect the network data, but I will discuss some general principles based on probabilistic symmetries such as permutation invariance. Just like other probabilistic invariances, the distributional invariance to permuting indices of a matrix of interactions implies a representation theorem (the Aldous-Hoover theorem). This representation is in terms of a graph limit function, or graphon. I will discuss the representation, how to make inferences based on this representation, what to do if distributional permutation invariance does not hold, and what to do if we have additional information such as time stamp of interactions, multiple interactions or additional covariate data.
Björn Sprungk - Noise-level robust sampling and Bayesian inference on the sphere
We consider two topics in this talk: the first is related to sampling from concentrated posterior distributions arising in Bayesian inference with informative data. Although a desirable situation from an inference perspective concentrated posteriors pose a computational challenge for many sampling algorithms. We present results regarding Markov chain Monte Carlo methods based on the Laplace approximation which show a statistical efficiency independent of the concentration of the posterior under suitable assumptions.
The second topic again considers the construction of Markov chain Monte Carlo algorithms but this time for dimension-independent sampling from posterior measures defined on a high-dimensional sphere as occuring in Bayesian density estimation. Both topics are related by the concept of pushforward Markov kernels.