On a generalization of the preconditioned Crank-Nicolson Metropolis algorithm

TL;DR

This paper introduces a generalized pCN Metropolis algorithm that adapts to the target measure, demonstrating dimension-independent performance and inheriting geometric ergodicity, with applications in Bayesian inverse problems.

Contribution

A novel generalized pCN proposal that incorporates measure information, improving sampling efficiency in high-dimensional spaces.

Findings

Performance independent of state space dimension

Inherits geometric ergodicity from pCN

Effective in Bayesian inverse problems

Abstract

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 of the measure of interest. A numerical simulation of a Bayesian inverse problem indicates that a Metropolis algorithm with such a proposal performs independent 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 ergodicity from the Metropolis algorithm with pCN proposal.