Abstract: Data assimilation in high-dimensional chaotic systems is a recurring challenge across disciplines, from meteorology to aerospace engineering. In the Bayesian filtering problem, the posterior distribution of the state of a dynamical system given partial, noisy, past observations is updated recursively. Despite theoretical advances on these Bayesian updates, dynamical information of the underlying chaotic model that can be inferred from simulation data have not been rigorously exploited in Bayesian filtering algorithms. In this talk, we aim to connect the concentration properties of the filtering recursion to observations on the unstable manifold. Further, we explore how this connection can be exploited to speed up numerical approximations of the filtering distributions. Dimension reduction techniques proposed in the fully Bayesian setting parallel the extensive development of rank reduction using the unstable subspace in algorithms based on Kalman updates, such as the extended Kalman filter.