A hybrid adaptive MCMC algorithm in function spaces

TL;DR

This paper introduces a hybrid adaptive MCMC algorithm for function spaces that combines adaptive Metropolis and pCN methods, improving efficiency and maintaining ergodicity, with demonstrated competitive performance.

Contribution

It develops a novel hybrid adaptive MCMC algorithm that enhances pCN efficiency in function space Bayesian inference while ensuring ergodicity.

Findings

The hybrid algorithm maintains ergodicity under certain conditions.

Numerical examples show competitive performance with existing methods.

The method adapts in finite-dimensional subspaces for improved sampling efficiency.

Abstract

The preconditioned Crank-Nicolson (pCN) method is a Markov Chain Monte Carlo (MCMC) scheme, specifically designed to perform Bayesian inferences in function spaces. Unlike many standard MCMC algorithms, the pCN method can preserve the sampling efficiency under the mesh refinement, a property referred to as being dimension independent. In this work we consider an adaptive strategy to further improve the efficiency of pCN. In particular we develop a hybrid adaptive MCMC method: the algorithm performs an adaptive Metropolis scheme in a chosen finite dimensional subspace, and a standard pCN algorithm in the complement space of the chosen subspace. We show that the proposed algorithm satisfies certain important ergodicity conditions. Finally with numerical examples we demonstrate that the proposed method has competitive performance with existing adaptive algorithms.