Week 24.03.2025 – 29.03.2025

Motivated by statistical applications, I will illustrate aspects of excursion theory for the Wright--Fisher diffusion with recurrent mutation, a fundamental model playing a central role in population genetics. The structure is intermediate between the classical excursion theory, where all excursions begin and end at a single point, and the more general approach considering excursions of processes from general sets. Since the Wright--Fisher diffusion has two boundary points, it is natural to construct excursions which start from a specified boundary point, and end at one of two boundary points which determine the next starting point. In order to do this we study the killed Wright--Fisher diffusion, which is sent to a cemetery state whenever it hits either endpoint. Several identities for excursion measures and hitting time distributions will be described both via special function theory and via the coalescent dual.

Deterministic mathematical models, such as those specified via differential equations, are a powerful tool to communicate scientific insight. However, such models are necessarily simplified descriptions of the real world. Generalised Bayesian methodologies have been proposed for inference with misspecified models, but these are typically associated with vanishing parameter uncertainty as more data are observed. In the context of a misspecified deterministic mathematical model, this has the undesirable consequence that posterior predictions become deterministic and certain, while being incorrect. Taking this observation as a starting point, we propose Prediction-Centric Uncertainty Quantification, where a mixture distribution based on the deterministic model confers improved uncertainty quantification in the predictive context. Computation of the mixing distribution will be cast as a (regularised) gradient flow of the maximum mean discrepancy (MMD), enabling consistent numerical approximations to be obtained. The idea will be illustrated using a model of protein signalling in cell biology.