# Bayes Filter

tags
Gaussian Filter

## Notation

$$\eta$$
normalizing constant, to make probability distribution sum to 1.
$$\text{bel}(t) = p(x_t | z_{1:t}, u_{1:t})$$
posterior probabilities over state variables conditioned on available data
$$\overline{\text{bel}}(t) = p(x_t | z_{1:t-1}, u_{1:t})$$
belief taken before incorporating the measurement $$z_t$$

$$\overline{\text{bel}}(t)$$ often called the prediction in Bayes filtering. Computing $$\text{bel}(t)$$ from $$\overline{\text{bel}}(t)$$ is called correction or the measurement update.

## Algorithm

\begin{algorithm} \caption{Bayes Filtering} \label{bayes_filter} \begin{algorithmic}[1] \Procedure{BayesFilter}{$\text{bel}(x_{t-1}), u_t, z_t$} \ForAll{$x_t$} \State $\overline{\text{bel}}(t) = \int p(x_t | u_t, x_{t-1}) \text{bel}(x_{t-1}) dx$ \State $\text{bel}(t) = \eta p(z_t | x_t)\overline{\text{bel}}(t) (x_t)$ \EndFor \State \Return $bel(x_t)$ \EndProcedure \end{algorithmic} \end{algorithm}

## Details

Mathematical derivation makes the Markovian Assumption.

Exact techniques for belief calculation are reserved for specialized cases. In most scenarios, these beliefs have to be approximated, and these approximations have important ramifications on the complexity of the algorithm.