Kalman smoother (RTS / fixed-interval)

A smoother estimates each hidden state using observations from both before and after that time, unlike a filter, which uses only the past. The Rauch–Tung–Striebel algorithm does this efficiently for linear-Gaussian models with a forward Kalman pass followed by a backward recursion that corrects each filtered estimate using information that arrived later. Because it exploits future data, the smoothed estimate is always at least as accurate as the filtered one and is markedly smoother.

The catch is that using future data makes smoothing acausal, so it cannot run in a real-time control loop — the decoder cannot wait for observations it has not yet received. Smoothers are therefore offline tools in BCI: reconstructing trajectories for analysis, computing the sufficient statistics needed to learn state-space parameters (they are the E-step of the EM algorithm), and providing a best-case benchmark of how much information the data contain relative to what the causal filter achieves.