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Kalman Filter

A recursive algorithm that computes the optimal estimate of an unobserved, continuously evolving state from a stream of noisy observations, for the special case where the dynamics and noise are linear and Gaussian.

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In Plain English

The Kalman filter is the continuous-state cousin of the forward algorithm for hidden Markov models: instead of a discrete hidden state, it tracks a continuous quantity (a rocket's position and velocity, an asset's underlying trend) that evolves over time and is only ever observed through noisy measurements. At every new observation, the filter does two things: predicts where the state should be based on its known dynamics, then updates that prediction by blending it with the new noisy measurement, weighted by how much to trust each source. Under the assumption that everything involved is linear and Gaussian, this simple predict-update cycle is provably the best possible estimator, not just a good heuristic.

Definition

The Kalman filter solves the filtering problem for a linear-Gaussian state-space model: a state equation $X_{t} = F X_{t - 1} + w_{t}$ , $w_{t} \sim N (0, Q)$ , and an observation equation $Y_{t} = H X_{t} + v_{t}$ , $v_{t} \sim N (0, R)$ , with $w_{t}$ and $v_{t}$ independent Gaussian noise. Given a prior estimate of the state's mean and covariance, the filter alternates a predict step, projecting the state estimate forward using the state equation, and an update step, correcting that prediction using the new observation $Y_{t}$ via the Kalman gain $K_{t}$ , which optimally weights the prediction against the new measurement based on their relative uncertainties. Because both steps preserve Gaussianity exactly (linear transformations of Gaussians are Gaussian), the filter's output at every step is the exact posterior distribution of the state given all observations so far, making it the minimum-mean-squared-error estimator, provably optimal, not merely a reasonable approximation, under the linear-Gaussian assumptions.

Formula

X_{t} = F X_{t - 1} + w_{t}, Y_{t} = H X_{t} + v_{t}, w_{t} \sim N (0, Q), v_{t} \sim N (0, R)

Notation

F, H

—State transition matrix and observation matrix

Q, R

—Covariance matrices of the process (state) noise and observation noise

K_{t}

—Kalman gain: optimal weighting of the new observation against the prior prediction

Properties

Developed by Rudolf Kalman in 1960, the filter's most famous early application was NASA's Apollo program guidance computer, which used it to fuse noisy sensor readings into a precise real-time trajectory estimate, cementing its reputation as one of the most consequential algorithms in applied engineering history.
The Kalman filter is provably optimal only under its stated assumptions, linear dynamics, linear observations, and Gaussian noise, when any of these fail, extensions like the extended Kalman filter (locally linearizing nonlinear dynamics) or the unscented Kalman filter (a more accurate nonlinear approximation) are used instead, or a particle filter for severely non-Gaussian problems.
The filter is the continuous-state, linear-Gaussian sibling of the hidden Markov model's forward algorithm, both solve exactly the same abstract filtering problem, estimate the current hidden state given observations so far, just for different classes of state-space model.

At a Glance

Applies tolinear-Gaussian state-space models

Two-step cyclepredict (project state forward) then update (blend with new observation)

Optimalitythe exact minimum-mean-squared-error estimator, not an approximation, in this setting

Beyond linear-Gaussiannonlinear extensions (extended, unscented Kalman filter) or particle filters

Last updated September 8, 2026← Back to the Dictionary