# Gaussian Filter

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.