Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods

In this chapter we overview a Bayesian approach to a wide range of signal processing problems in which the goal is to find the signal, which is a solution of an ordinary or stochastic differential equation, given noisy observations of its solution. In the case of ordinary differential equations (ODEs) this gives rise to a finite dimensional probability measure for the initial condition, which then determines the measure on the signal. In the case of stochastic differential equations (SDEs) the measure is infinite dimensional, on the signal itself, a time-dependent solution of the SDE. We derive the posterior measure for these problems, applying the ideas to ODEs and SDEs, with discrete or continuous observations, and with coloured or white noise. We highlight common structure inherent in all of the problems, namely that the posterior measure is absolutely continuous with respect to a Gaussian prior. This structure leads naturally to the study of Langevin equations which are invariant for the posterior measure and we highlight the theory and open questions relating to these S(P)DEs. We then describe the construction of effective Metropolis-based sampling methods for the posterior measure, based on proposals which can be interpreted as approximations of the Langevin equation.