Kalman Filter
Kalman Filter (Linear-Gaussian State-Space Filter) · Also known as: linear quadratic estimator, LQE, Kalman-Bucy filter, optimal recursive filter
The Kalman filter is an optimal recursive algorithm for estimating the hidden state of a linear dynamical system from noisy measurements. At each time step it alternates between a prediction step — projecting the state forward using the system model — and an update step that corrects the prediction with the new observation, producing minimum-variance state estimates and their uncertainty in real time.
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When to use it
Use the Kalman filter when your system is linear, noise is Gaussian, and you need real-time or recursive state estimation from sequential measurements — classic examples include sensor fusion, navigation (GPS/IMU), target tracking, and econometric state-space models. It is the optimal filter under these assumptions and runs in closed form with no sampling. Do not use it when the system dynamics or observation model are nonlinear (use the Extended or Unscented Kalman filter, or a particle filter instead), when noise distributions are heavy-tailed or multimodal, or when the state-space dimension is so large that matrix inversions become prohibitive.
Strengths & limitations
- Computationally efficient and recursive: each update is O(n^3) in state dimension and requires no storage of past data.
- Optimal (minimum mean-squared-error) linear estimator under Gaussian noise assumptions.
- Naturally provides uncertainty estimates for every state estimate via the covariance matrix.
- Handles missing observations gracefully by simply running the prediction step without an update.
- Well-understood theory with decades of engineering validation across navigation, signal processing, and finance.
- Assumes linear state-transition and observation models — violated by most real-world nonlinear systems.
- Requires Gaussian noise; heavy-tailed or multimodal noise corrupts optimality guarantees.
- Performance depends critically on correct specification of Q (process noise) and R (measurement noise) covariances, which are often unknown and must be tuned or estimated.
- Can diverge if the model is misspecified or initial covariance is poorly chosen.
Frequently asked
What is the difference between the Kalman filter and a particle filter?
The Kalman filter is exact and closed-form but restricted to linear-Gaussian systems. A particle filter (sequential Monte Carlo) uses a cloud of weighted samples to represent the posterior and can handle nonlinear, non-Gaussian models, at the cost of much greater computation and variance.
How do I choose Q and R?
R (measurement noise) is often available from sensor specifications or calibration data. Q (process noise) is harder — start from physical reasoning about unmodeled disturbances, then refine using normalized innovation squared (NIS) tests or expectation-maximization (EM) to estimate Q from data.
What is the Extended Kalman Filter (EKF)?
The EKF linearizes nonlinear state and observation functions around the current state estimate using first-order Taylor expansion, then applies the standard Kalman update. It is widely used in robotics and navigation but can diverge when nonlinearity is strong. The Unscented Kalman Filter (UKF) uses a deterministic sigma-point approximation and is generally more accurate.
Is the Kalman filter Bayesian?
Yes. It is the exact solution to recursive Bayesian filtering for linear-Gaussian systems. The prediction step computes the Chapman-Kolmogorov prior, and the update step applies Bayes' rule; because both prior and likelihood are Gaussian, the posterior is also Gaussian and is computed exactly.
Can the Kalman filter handle missing observations?
Yes. When a measurement is missing at time t, simply skip the update step and carry the predicted distribution forward as the posterior. This is one of the practical advantages of the recursive Bayesian framework.
Sources
- Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1), 35-45. DOI: 10.1115/1.3662552 ↗
- Welch, G. & Bishop, G. (2006). An Introduction to the Kalman Filter. University of North Carolina at Chapel Hill, Technical Report TR 95-041. link ↗
How to cite this page
ScholarGate. (2026, June 3). Kalman Filter (Linear-Gaussian State-Space Filter). ScholarGate. https://scholargate.app/en/bayesian/kalman-filter
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Bayesian RegressionBayesian↔ compare
- Dynamic Bayesian NetworkBayesian↔ compare
- Extended Kalman FilterControl Theory↔ compare
- Particle FilterBayesian↔ compare
- Sequential Monte CarloBayesian↔ compare