> For the complete documentation index, see [llms.txt](https://calvinfeng.gitbook.io/probabilistic-robotics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://calvinfeng.gitbook.io/probabilistic-robotics/basics/gaussian-filters.md).

# Gaussian Filters

We are going to introduce an important family of recursive state estimators, collectively called **Gaussian filters**. Gaussian techniques all share the basic idea that beliefs are represented by multivariate normal distributions.

$$
p(x) = det(2\pi\Sigma)^{-0.5} \exp\left(\frac{-1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)\right)
$$

The density over the variable $$x$$ is characterized by two sets of parameters. The mean $$\mu$$ and covariance matrix $$\Sigma$$. Gaussians are unimodal; they possess a single maximum. This may be suitable for some problems but may not be appropriate for problems that exist many distinct hypotheses.

## Parameterization

The parameterization of a Gaussian by its mean and covariance is called the *moments parametrization*. This is because the mean and covariance are the first and second moments of a probability distribution. There is an alternative parameterization called *canonical parameterization* which will be discussed later.