The Hilbert projective metric is a generalisation, due to Hilbert, of the classic Cayley—Klein distance in hyperbolic geometry. In 1957 Birkhoff proved that, under the Hilbert metric, positive linear operators on positive convex cones contract. While this result has found applications in other branches of analysis, it has generally been overlooked by the probabilistic community, despite providing, for example, a straightforward approach to the study of the stability of Markov chains. In this talk I will give an overview of the Hilbert metric and its properties, with a particular focus on its merits in applied probability theory. Starting in greater generality by working on locally convex topological vector spaces, I will give a definition of the Hilbert metric using duality, and provide a proof of (a slighter stronger version of) Birkhoff’s contraction result. I will then look specifically at the space of probability measures equipped with the Hilbert metric, and explore its structure, carrying on a comparison with other metrics on the side. Finally, as an application, I will introduce the stochastic filtering problem, and how the Hilbert metric can be effectively used to prove stability of the filtering equations.

This talk is based on joint work with Sam Cohen (Oxford).