A Derivation of the Kalman Filter

The Kalman Filter is one of the more useful tools in data science, but while there are a lot of well-written descriptions of the Bayesian tracking technique available online and in technical books/articles, for some reason it’s hard to find a simple derivation of the Kalman Filter from first principles.

In this short note, I show how 1-dimensional Kalman Filter comes about from the properties of Bayes Rule, Fourier transform, and Gaussian distributions.

A Derivation of the Kalman Filter

The generalisation of the proof to multi-dimensional Kalman Filter can be done in a similar way.

 

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