Kalman Filter

The Kalman filter is a recursive algorithm that estimates the state of a dynamical system from a series of noisy measurements, and it is secretly a Bayesian updater in engineering clothing. Developed by Rudolf Kalman in 1960, the filter maintains a probability distribution over the system's state, predicts how that distribution evolves under the system's dynamics, and corrects the prediction using incoming observations weighted by their reliability. Under linear-Gaussian assumptions, the Kalman filter is the optimal estimator in the minimum-mean-square-error sense — but its deeper significance is that it demonstrates how Bayesian reasoning can run in real time, cycle after cycle, without ever requiring the complete data history to be stored or reprocessed. The filter is the computational backbone of guidance systems from the Apollo program to modern autonomous vehicles, and its architecture — predict, observe, correct — is the template for virtually all sequential Bayesian inference.