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.