  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.
