Interactive element. The Kalman filter's position update relies on three primary variables: the prior uncertainty (yellow), the observation error (blue), and the observation value (green). Adapt these values by dragging the sliders and observe how the compromise solution (the posterior; red) changes in response.

Gaussian inference

In a nutshell, Bayes' Theorem realizes two steps:

This operation can rarely be solved in closed form. However, when both the prior and the likelihood are Gaussian, the joint pdf p(x,y) is always (multivariate) Gaussian. This permits a closed-form solution. Interact with the element to the left, and observe how the posterior solution changes (and what remains unchanged) as you adjust the following three variables: