Bayesian inference for diffusion processes

The Bayesian approach to inverse problems is often valued for its regularizing properties and a way in which quantification of uncertainty is built-in directly into this paradigm. In recent decades, fueled by the increasing availability of the computing power, it has seen a rapid increase in popularity and an explosion of new applications.

The main point of this talk is to show how this framework can be used to estimate unknown parameters of a stochastic differential equation given discrete-time observations of the underlying process---a setup often encountered in neuroscience, physics, population dynamics and many other sciences. One of the main difficulties afflicting successful employment of the Bayesian approach to solving such problems is the necessity to simulate diffusion processes conditioned on their end-points---a well-studied problem that nonetheless does not have a universally robust and efficient solution. In this talk I will present some of the recent advances in simulation of conditioned diffusions and discuss some of the algorithmic difficulties in trying to extend those methodologies to more elaborate processes.