Bayesian approach to inverse problems for functions with variable index Besov prior

We adopt Bayesian approach to consider the inverse problem of estimate a function from noisy observations. One important component of this approach is the prior measure. Total variation prior has been proved with no discretization invariant property, so Besov prior has been proposed recently. Different prior measures usually connect to different regularization terms. Variable index TV, variable index Besov regularization terms have been proposed in image analysis, however, there are no such prior measure in Bayesian theory. So in this paper, we propose a variable index Besov prior measure which is a Non-Guassian measure. Based on the variable index Besov prior measure, we build the Bayesian inverse theory. Then applying our theory to integer and fractional order backward diffusion problems. Although there are many researches about fractional order backward diffusion problems, we firstly apply Bayesian inverse theory to this problem which provide an opportunity to quantify the uncertainties for this problem.