Treffer: Avariational Approach to Nonlinear Estimation

We consider estimation problems, in which the estimand, X, and observation, Y , take values in measurable spaces. Regular conditional versions of the forward and inverse Bayes formula have variational characterisations involving the minimisation of an apparent information, and the maximisation of a compatible information. These both have natural information theoretic interpretations, according to which Bayes' formula and its inverse are optimal information processors. The variational characterisation of the forward formula has the same form as that of Gibbs measures in statistical mechanics. The special case in which X and Y are diffusion processes governed by stochastic differential equations is examined in detail. The minimisation of apparent information is shown then to involve a stochastic optimal control problem for the signal process, with cost that is quadratic in both the control and observation fit. This leads to a differential formula for the pathwise nonlinear interpolator. Local versions of the variational characterisations are developed, which quantify information flow in the estimators. In this context, the information conserving property of bayesian estimators coincides with the Davis-Varaiya martingale stochastic dynamic programming principle.