Learning state-space parameters (EM / maximum likelihood)

The matrices of a state-space decoder — the dynamics A and W and the observation model C and Q — must be fit to data before the filter can run. When the state is observed during calibration (as in supervised BCI calibration, where the user tracks a known target so the intended kinematics are treated as ground truth), these can be estimated in closed form by least squares. When the state is latent or only partially known, the expectation–maximization algorithm alternates between a smoother that infers the state distribution given current parameters (E-step) and a closed-form update of the parameters given that distribution (M-step), climbing the likelihood to a local optimum.

In BCI this is how encoding models and dynamics priors are learned and re-learned, and it is where the honest difficulties live: the likelihood is non-convex so results depend on initialization; short calibration blocks give noisy estimates that overfit; and the ground-truth intention used in supervised fitting is itself only an assumption (which is exactly what ReFIT tries to repair). Regularization and shrinkage of the estimated covariances are usually essential for the fitted filter to behave well online.