Conclusion

The sequential estimation of a hidden process from an observed process continues to be the subject of an abundant scientific literature. Wiener and Kalman laid the foundations. Following on from Wiener's work, Kalman proposed a recursive estimator based on a dynamic linear state model, taking advantage of all past observations. Depending on the nature of the noise, the Kalman filter is a MAP estimator (Gaussian noise and initial state) or an EQMM estimator with linearity constraint in the non-Gaussian case.

If the state model is non-linear, the most immediate solution is to linearize it. The imprecision introduced is reflected in the degraded performance of the extended Kalman filter relative to MAP or EQMM criteria. In the case where the dynamic state model is non-linear and/or the noise is non-additive, sequential Monte-Carlo methods are used.