Recursive Bayesian estimation (Bayes filter)

Recursive Bayesian estimation maintains the full posterior distribution over the hidden state given all observations so far, updating it one measurement at a time in two alternating steps. The prediction step propagates the previous posterior through the dynamics (the Chapman–Kolmogorov equation), spreading probability mass and typically increasing uncertainty. The update step multiplies this prediction by the observation likelihood and renormalizes (Bayes' rule), sharpening the estimate toward what the new data support.

This predict–update recursion is the common backbone of every filter in this field. The variants differ only in how they represent and propagate the posterior: the Kalman filter assumes it stays Gaussian and tracks a mean and covariance in closed form; the particle filter represents it by weighted samples; the point-process filter keeps a Gaussian approximation but with a spike-train likelihood. Working recursively means the estimate is available in real time and memory does not grow with the length of the session, which is essential for online, closed-loop control.