Diffusion limits for MCMC paths

We examine complexity and optimal tuning of MCMC algorithms in high dimensions. Earlier works in the literature have shown that the step of Random-Walk Metropolis should be scaled as $1/n$ ($n$ being the dimensionality index); for the (Metropolised) Langevin algorithm the scaling is $1/n^{1/3}$. Both results have been theoretically justified in the simplified scenario of iid targets. Under these scalings MCMC trajectories converge to analytically identifiable diffusions. Such a limit allows for a complete characterisation of the algorithms and the straightforward specification of optimality criteria, via the identification of the best acceptance probability that practitioners should try to achieve.

In recent research, we have extended this framework to complex, realistic models, including conditioned diffusion models and inverse problems for the Navier-Stokes equation. We have established optimality criteria for MCMC algorithms applied on such structures and identified algorithmic complexity and best acceptance probability.
In these cases, MCMC trajectories are shown to converge to infinite dimensional Stochastic Differential Equations. We have also consider the so-called hybrid Monte-Carlo MCMC algorithm used by physicists in molecular dynamics and other applications and studied its performance in high dimensions.