The Bernstein-von Mises theorem for non-regular generalised linear inverse problems
We consider a broad class of nonlinear statistical inverse problems from a Bayesian perspective. This provides a flexible and interpretable framework for their analysis, but it is important to understand the relationship between the chosen Bayesian model and the resulting solution, especially in the ill-posed case where in the absence of prior information the solution is not unique. Following earlier work about consistency of the posterior distribution of the reconstruction, we obtain approximations to the posterior distribution in the form of a Bernstein–von Mises theorem for nonregular Bayesian models. Emission tomography is taken as a canonical example for study, but our results hold for a wider class of generalised linear models with constraints.
Let X and Y be two inflnite dimensional real or complex Banach spaces. In this note we determine the forms of surjective additive maps ` : L(X) ! L(Y ) preserving the kernel's dimension or the range's codimension. As consequence, we establish that ` : L(X) ! L(X) preserves the kernel (respectively, the range) if and only if there exists an invertible operator A 2 L(X) such that `(T) = AT (respectively, `(T) = TA) for all T 2 L(X).