DISCRETIZATION-INVARIANT BAYESIAN INVERSION AND BESOV SPACE PRIORS

Bayesian solution of an inverse problem for indirect measurement M = AU + epsilon is considered, where U is a function on a domain of R(d). Here A is a smoothing linear operator and epsilon is Gaussian white noise. The data is a realization m(k) of the random variable M(k) = P(k)AU + P(k)epsilon, where P(k) is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as U(n) = T(n)U, where T(n) is a finite dimensional projection, leading to the computational measurement model M(kn) = P(k)AU(n) + P(k)epsilon. Bayes formula gives then the posterior distribution pi(kn)(u(n) vertical bar m(kn)) similar to Pi(n)(u(n)) exp(-1/2 parallel to m(kn) - P(k)Au(n) parallel to(2)(2)) in R(d), and the mean u(kn) := integral u(n) pi(kn)(u(n) vertical bar m(k)) du(n) is considered as the reconstruction of U. We discuss a systematic way of choosing prior distributions Pi(n) for all n >= n(0) > 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 Pi(n) represent the same a priori information for all n and that the mean u(kn) converges to a limit estimate as k, n -> infinity. 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)(1) prior is related to penalizing the l(1) norm of the wavelet coefficients of U.