Kalman Filter

What is the Kalman filter?

Answer

The Kalman filter is a filtering algorithm used to track/estimate some latent variable from noisy observations.

The idea is that at each step we have a model-based prediction for the latent variable, with some uncertainty/variance attached to it, and we also receive an observation, which is noisy and has its own variance. The Kalman filter combines these two sources of information using the Kalman gain.

Intuitively the Kalman gain says:

K = \frac{\text{model uncertainty}} {\text{model uncertainty} + \text{observation noise}}

If the observation variance/noise is high relative to the model variance, then the observation is less reliable and the filter puts more weight on the model prediction.
If the model uncertainty/variance is high relative to the observation variance/noise, then the model is less reliable and the filter puts more weight on the observation.

So basically, the Kalman filter updates the estimate by balancing how much we trust the model against how much we trust the new observation.

x_{\text{new}}=x_{\text{model}}+K(\text{surprise})

where

\text{surprise}=y-Hx_{\text{model}},

meaning the actual observation minus what the observation would be given our prediction.

Here, $$H$$ maps the state $$x$$ into the predicted observation. In the scalar case, $$H$$ is the scalar that turns $$x$$ into the predicted observation.