  In this paper, we exploit the gradient flow structure of continuous-time
formulations of Bayesian inference in terms of their numerical time-stepping.
We focus on two particular examples, namely, the continuous-time ensemble
Kalman-Bucy filter and a particle discretisation of the Fokker-Planck equation
associated to Brownian dynamics. Both formulations can lead to stiff
differential equations which require special numerical methods for their
efficient numerical implementation. We compare discrete gradient methods to
alternative semi-implicit and other iterative implementations of the underlying
Bayesian inference problems.
