On a Generalization of the Preconditioned Crank–Nicolson Metropolis Algorithm

Metropolis algorithms for approximate sampling of probability measures on infinite dimensional Hilbert spaces are considered, and a generalization of the preconditioned Crank–Nicolson (pCN) proposal is introduced. The new proposal is able to incorporate information on the measure of interest. A numerical simulation of a Bayesian inverse problem indicates that a Metropolis algorithm with such a proposal performs independently of the state-space dimension and the variance of the observational noise. Moreover, a qualitative convergence result is provided by a comparison argument for spectral gaps. In particular, it is shown that the generalization inherits geometric convergence from the Metropolis algorithm with pCN proposal.