Inverse problems: A Bayesian perspective

ion is not needed. Inverse problems 531 Theorem 6.13. Two Gaussian measures μi = N (mi, Ci), i = 1, 2, on a Hilbert space H are either singular or equivalent. They are equivalent if and only if the following three conditions hold: (i) Im(C 1 ) = Im(C 1/2 2 ) := E, (ii) m1 −m2 ∈ E, (iii) the operator T := ( C−1/2 1 C 1/2 2 )( C−1/2 1 C 1/2 2 )∗ − I is Hilbert–Schmidt in E. In particular, choosing C1 = C2 we see that shifts in the mean give rise to equivalent Gaussian measures if and only if the shifts lie in the Cameron– Martin space E. It is of interest to characterize the Radon–Nikodym derivative arising from such shifts in the mean. Theorem 6.14. Consider two measures μi = N (mi, C), i = 1, 2, on Hilbert space H, where C has eigenbasis {φk, λk}k=1. Denote the Cameron– Martin space by E. If m1 −m2 ∈ E, then the Radon–Nikodym derivative is given by dμ1 dμ2 (x) = exp ( 〈m1 −m2, x−m2〉C − 1 2 ‖m1 −m2‖C )