Mathematical Colloquium Statistical Inference for infinite-dimensional dynamical systems

We study optimal statistical inference procedures for the states of time evolution phenomena occurring in `data assimilation’ or filtering problems. There it is a common practice to assign a Gaussian process prior on the initial condition of a dynamical system and to update it to a Bayesian posterior measure in the space of possible trajectories given a discrete sample of the process. In key applications the dynamics are non-linear, such as with 2D Navier-Stokes equations in geophysical sciences or reaction-diffusion equations in biochemistry. While Bayesian posterior distributions are widely computed by filtering or MCMC methods, little is known about the statistical behaviour of these posterior measures in non-linear settings. In this talk we will introduce a theoretical framework for such models and then present recent results, known as `Bernstein-von Mises theorems’, that show that the posterior measures are approximated in function space by the Gaussian laws of solutions to certain SPDEs that involve the inverse Fisher information of the underlying statistical model.