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

The preconditioned Crank-Nicolson (pCN) method is a MCMC algorithm for implementing the Bayesian inferences in function spaces. A remarkable feature of the algorithm is that, unlike many usual MCMC algorithms, which become arbitrary slow under the mesh refinement, the efficiency of the algorithm is dimension independent. In this work we develop an adaptive version of the pCN algorithm, where the proposal is adaptively improved based on the sample history. Under the chosen parametrization of the proposal distribution, the proposal parameters can be efficiently updated in our algorithm. We show that the resulting adaptive pCN algorithm is dimension independent and has the correct ergodicity properties. Finally we provide numerical examples to demonstrate the efficiency of the proposed algorithm.