An adaptive independence sampler MCMC algorithm for Bayesian inferences of functions

Many scientific and engineering problems require one to perform Bayesian inferences in function spaces, in which the unknowns are of infinite dimension. In such problems, many standard Markov chain Monte Carlo (MCMC) algorithms become arbitrarily slow under the mesh refinement, which is referred to as being dimension dependent. In this work we develop an independence sampler based MCMC method for the Bayesian inferences of functions. We represent the proposal distribution as a mixture of a finite number of specially parametrized Gaussian measures. We also design an efficient adaptive algorithm to adjust the parameter values of the mixtures from the previous samples. Finally we provide numerical examples to demonstrate the efficiency and robustness of the proposed method, even for problems with multimodal posterior distributions.