THE KALMAN FILTER reconstruct the true state from a stream of noisy guesses — predict, then correct
A Kalman filter reconstructs the hidden state of a system — a rocket's position, a phone's heading — from measurements that are all noisy and a model that is only approximate. Each step is two moves: predict the next state from the model, then update it by blending in the new measurement, weighted by a gain that trusts whichever is more certain. The result is the optimal estimate under Gaussian noise — less noisy than the measurements, less wrong than the model alone. It flew Apollo to the Moon and lives in every GPS and drone. Reconstruction as a running fusion: the true value, recovered from a stream that never shows it directly.
THE TECHNIQUE predict from model, update with measurement, gain = who to trust
A prior estimate and a noisy measurement of one quantity. The demo computes the Kalman gain and fuses them into the reconstructed state: live demo
HISTORY & CREDIT Rudolf Kálmán, 1960
“A noisy sensor gives a noisy answer.” — fuse it with a model and the past, weighted by certainty, and the estimate is better than either source. The true state is reconstructed from data that never shows it cleanly. cited
1960 · Rudolf Kálmán — the recursive optimal linear filter (Kalman–Bucy with Bucy, 1961); precursors in Thiele and Swerling. 1969 · flew the Apollo guidance computer. now · GPS, drones, robotics, finance, sensor fusion everywhere.
Predict from the model, correct with the measurement, and trust each by its certainty — the estimate beats both. The true state, reconstructed from a noisy stream. Kálmán 1960
RECOMMEND FOR I-13 state reconstructed by predict-update, computed
On the canonical compiler, a prior 5 (variance 2) and a measurement 6 (variance 2) give gain K=0.5 and a fused estimate 5.5, with reduced variance 1:
$ i13 run rec_kalman.i13 # predict + update
K = 0.5 -- gain: trust prior and measurement equally here
xpost = 5.5 -- the reconstructed state, between prior and measurement
Ppost = 1 -- and MORE certain than either (variance dropped 2 -> 1)
Recommend: the Kalman filter is reconstruction as running fusion, and i13 computes one step exactly: prior 5, measurement 6, gain K=P/(P+R)=0.5, fused estimate 5.5, variance dropping 2→1. The load-bearing structure is the certainty-weighting: the gain is derived so the estimate is optimal (minimum variance), and the result is more certain than either input — the recovery improves the state, not just averages it. It is the batch's estimation apex: where least squares (dart 305) reconstructs a static trend, Kalman reconstructs a moving one, correcting itself measurement by measurement. The true value recovered from a stream that never shows it.