Recursive Bayesian Smoothing with Gaussian Sums

Abstract Recursive Bayesian inference is well established; for Gaussian models, the Kalman filter and Rauch-Tung-Striebel (RTS) smoother provide closed-form recursions. For mixture models such as Gaussian sums, prediction and filtering remain tractable, but the standard fixed-lag smoothing recursion introduces a density division (a ratio of sums) and is not algebraically closed within the Gaussian-mixture family. As a result, smoothing is often implemented via two-filter decompositions, which do not yield a recursion directly in the smoothing marginals. We derive an explicit closed-form correction step for recursive Bayesian smoothing with Gaussian mixtures that is purely recursive in time and avoids ratios of Gaussian sums by expressing the smoothing correction as a finite sum of componentwise Gaussian fractions. Each component reduces to an RTS-like Gaussian correction, so the mixture recursion can be evaluated in closed form using standard linear-Gaussian algebra. The resulting correction rules provide closed-form expressions for all parameters of the smoothing mixture, enabling analytic propagation within a finite Gaussian sum. Since Gaussian mixtures are universal approximators, the result provides a broadly applicable analytic core for smoothing in nonlinear and multimodal settings.