Gaussian Filters is a tractable implementation of the Bayes filter (§bayes_filter) for continuous spaces.

Key Idea

Beliefs are represented by a multi-variate normal distribution.

\begin{equation} p(x) = \text{det}(2 \pi \Sigma)^{-\frac{1}{2}} \text{exp} \left( - \frac{1}{2} (x -\mu)^T \Sigma^{-1} (x- \mu) \right) \end{equation}

The density of variable \(x\) is characterized by mean \(\mu\) and covariance \(\Sigma\).

Ramifications

Since beliefs are represented by a multi-variate normal distribution, this means that beliefs are uni-modal. This is suitable for many tracking problems. However, this is a poor match for many global estimation problems with multiple hypotheses that should give rise to their own modes in the posterior.

Representations

moments representation
The Gaussian is represented by its mean and covariance (first and second moments)
canonical representation :

These representations have a bijective mapping, and are functionally equivalent, but give rise to different algorithms.

Using the moments representation gives rise to the Kalman Filter (§kalman_filter).