On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inferences
For researchers performing Bayesian inference on infinite-dimensional parameters, this work offers an adaptive variant of the pCN algorithm to improve sampling efficiency, though it is an incremental improvement.
The paper develops an adaptive version of the preconditioned Crank-Nicolson MCMC algorithm for infinite-dimensional Bayesian inference, which adjusts the proposal covariance based on sampling history to improve efficiency. Numerical examples demonstrate performance, but no concrete numbers are provided.
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.