Bayes Filter

The core idea of the application of Bayes' rule is that we can update our belief based on the new data. Here is a classic example in many robotics text books that illustrates the idea:

This example might be too simple. An example with multiple doors is more interesting for the thought process but we will leave it as an exercise.

Graph Model Representation

The Bayes Filter algorithm consists of two steps:

• prediction

• belief update

TODO

Bayes Filter Algorithm

\begin{algorithm}
    \renewcommand{\thealgorithm}{}
    \caption{\textbf{Bayes Filter}}\label{alg:cap}
    
    \begin{algorithmic}[1]
        \Function{BayesFilter}{$bel(x_{t-1}), u_t, z_t$}
        \ForAll{$x_t$}
        \State $\overline{bel}(x_t) = \int{p(x_t|u_t, x_{t-1})bel(x_{t-1})dx_{t-1}}$
        \State $bel(x_t) = \eta{}p(z_t|x_t)\overline{bel}(x_t)$
        \EndFor
        \State \textbf{return} $bel(x_t)$
        \EndFunction
        
    \end{algorithmic}
\end{algorithm}