Discretization-invariant Bayesian Inversion and Besov Space Priors

Bayesian solution of an inverse problem for indirect measurement M = AU + E is considered, where U is a function on a domain of R. Here A is a smoothing linear operator and E is Gaussian white noise. The data is a realization mk of the random variable Mk = PkAU + PkE, where Pk is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as Un = TnU , where Tn is a finite dimensional projection, leading to the computational measurement model Mkn = PkAUn + PkE. Bayes formula gives then the posterior distribution πkn(un |mkn) ∼ Πn(un) exp(− 1 2 ‖mkn − PkAun‖ 2 2) in R , and the mean ukn := R un πkn(un |mk) dun is considered as the reconstruction of U . We discuss a systematic way of choosing prior distributions Πn for all n ≥ n0 > 0 by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Πn represent the same a priori information for all n and that the mean ukn converges to a limit estimate as k, n → ∞. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with B 11 prior is related to penalizing the l norm of the wavelet coefficients of U .