  Many scientific and engineering problems require 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 arbitrary 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 infinite dimensional Bayesian inferences. We represent the
proposal distribution as a mixture of a finite number of specially parametrized
Gaussian measures. We show that under the chosen parametrization, the resulting
MCMC algorithm is dimension independent. 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.
