# Bayes 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.

Approximations requires trading off:

Computational Efficiency
Linear Gaussian approximations can calculate beliefs in time polynomial to the state space. Other approximations may be exponential. Particle filter techniques have the any-time characteristic, allowing them to trade off accuracy with computational efficiency.
Accuracy
Linear Gaussian approximations are limited to unimodal distributions, whereas histogram approximations are multi-modal, but have limited accuracy. Particle representations can approximate a wide array of distributions, but large number of particles may be required for high accuracy.
Ease of implementation
Difficulty in implementation typically arises from the form of the transition and measurement probabilities (see §robotics_probabilistic_generative_laws). Particle representations lend themselves to simple implementations for complex non-linear systems.

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