Hidden Markov model decoder (discrete state)

When the quantity to decode is discrete rather than continuous — a task state, a grasp type, a click versus no-click, or a phoneme in a sequence — the natural state-space model is a hidden Markov model: a small set of discrete states with a transition matrix as the dynamics prior and a per-state emission distribution as the observation model. The recursive Bayesian update becomes the forward algorithm, which maintains a probability over states in real time; for offline best-path decoding the Viterbi algorithm is used, and forward–backward provides smoothed posteriors and the statistics needed to train the model.

HMM decoders are workhorses for the discrete side of BCI: detecting movement onset or state, gating a continuous decoder on or off, sleep and event staging, and, as a component, imposing sequence structure in speech and handwriting decoding where phoneme or character transitions are constrained. The transition matrix acts as a temporal prior in exactly the way the dynamics matrix does for a Kalman filter, and the same caveats apply — the Markov and stationarity assumptions are approximations, and a mis-estimated emission model bounds accuracy.