On the Analytic Solution to Recursive Smoothing in Mixture Models

Mixture distributions can be used as universal approximators for arbitrary probability densities. Within the framework of Bayesian inference based on state observation models, the usage of mixtures has the potential to significantly improve the marginal distributions obtained by fundamental inference tasks such as predicting, filtering and smoothing. The inference formulas are of recursive nature and have been well known for decades, except for mixture smoothing. The classic approach of incorporating the mixture assumptions into recursive formulations works for predicting and filtering, but does not work out for smoothing, which remained precarious and was even assumed to be infeasible. This article tackles this current state and provides, to our knowledge, the first proof of an analytic solution for recursive smoothing with mixture distributions. The main idea is to decompose the smoothing marginal into a non-recursive two-filter formulation, incorporate the mixture assumptions, and reformulate the mixture components into a recursive fixed-lag entanglement.