Durbin and Koopman (2002) derive an efficient smoothing algorithm for the Kalman filter as well as a neat sampling scheme for drawing from the posterior density of the states, conditional on all available observations. In this note, I restate key elements of both, using a slightly different notation for the state space system, than in the original paper.

The smoother is based on backwards-recursive projections on innovations derived from the Kalman filter's forward recursion: