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.