Complementary models and smoothing

The authors apply the concept of complementary models introduced by Weinert and Desai (1983) to derive forwards and backwards Markovian models for the smoothing error process. By exploring the structure of the complementary models they show that, under certain restrictions, only two simple structured models exist, one that runs forwards in time and another than runs backwards in time. The forwards complementary model leads to a forwards Rauch-Tung-Striebel (RTS) smoothing formula and to a backwards Markovian model for the error, whereas the backwards model leads to a backwards RTS formula and to a forwards error model. The two models for the smoothing error can be derived one from the other by a forwards-backwards transformation that preserves the sample paths. Finally, by using a combination of the two complementary models another proof of the two-filter smoothing formula is given