  An evolutionary form of a generalized Bayesian update method, which is
strictly derivative- free yet directed through an additive update term based
purely on the statistical moments of the design variables, is proposed for
nonlinear inverse problems in general and applied in particular to an optical
imaging problem, the ultrasound modulated optical tomography (UMOT). The
additive update term, which bypasses most pitfalls of a conventional weight-
based Bayesian update, results from a change of measures aimed at driving
appropriately derived observation-prediction error terms or increments of cost
functionals to zero-mean Brownian martingales. This constitutes a novel
characterization corresponding to the extremization of the cost functional(s),
where the design unknowns are represented as diffusion processes evolving with
respect to a continuously parameterized iteration variable. This leads to a
recursive prediction-update algorithm to implement the search. The scheme
offers freedom from sample degeneracy and the accompanying divergence of the
conventional weight-based Bayesian update schemes. We obtain the order of
convergence of the conditioned process and also establish that the solutions
are stable against tolerable variations in the regularizing noise terms, even
as the original inverse problem remains severely ill-posed. Numerical evidence
on solutions to the UMOT problem also confirms substantive improvements in the
reconstruction efficacy through the proposed method vis-\`a- vis a Gauss-Newton
approach, especially where the regularized quasi-Newton direction has low
sensitivity to variations in the design unknowns.
