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 are shown to have dual variational characterizations involving the minimization of apparent information and the maximization of compatible information. These both have natural information-theoretic interpretations, according to which Bayes' formula and its inverse are optimal information processors. The variational characterization 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 minimization of apparent information can then be formulated as a stochastic optimal control problem, with cost that is quadratic in both the control and observation fit. The dual problem can be formulated in terms of infinite-dimensional deterministic optimal control. Local versions of the variational characterizations 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.