On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inferences

TL;DR

This paper introduces an adaptive preconditioned Crank-Nicolson MCMC algorithm designed for efficient Bayesian inference in infinite-dimensional spaces, addressing the slow convergence issues of traditional methods.

Contribution

The paper develops an adaptive pCN algorithm that adjusts the covariance operator based on sampling history, ensuring ergodicity and improved efficiency in infinite-dimensional Bayesian problems.

Findings

The adaptive pCN algorithm satisfies ergodicity under mild conditions.

Numerical examples demonstrate enhanced sampling efficiency.

The method effectively handles high-dimensional Bayesian inference problems.

Abstract

Many scientific and engineering problems require to perform Bayesian inferences for unknowns of infinite dimension. In such problems, many standard Markov Chain Monte Carlo (MCMC) algorithms become arbitrary slow under the mesh refinement, which is referred to as being dimension dependent. To this end, a family of dimensional independent MCMC algorithms, known as the preconditioned Crank-Nicolson (pCN) methods, were proposed to sample the infinite dimensional parameters. In this work we develop an adaptive version of the pCN algorithm, where the covariance operator of the proposal distribution is adjusted based on sampling history to improve the simulation efficiency. We show that the proposed algorithm satisfies an important ergodicity condition under some mild assumptions. Finally we provide numerical examples to demonstrate the performance of the proposed method.