Bayes Filter

A Bayes filter abandons the idea of knowing a single, exact state. Instead, it tracks a Belief distribution—a probability density function representing how likely every possible state is, given all past sensor readings and control inputs.

1. Predict (The Prior):

You integrate across all possible previous states, applying your physical motion model ($p(x_t\mid x_{t-1},u_t)$).

$\overline{bel}(x_t)=\int p(x_t\mid x_{t-1},u_t)bel(x_{t-1})d{x}_{t-1}$

2. Update (The Posterior):

You multiply that predicted belief by the likelihood of getting your current sensor reading ($p(z_t\mid x_t)$), and normalize it with a constant $\eta$.

$bel(x_t)=\eta p(z_t\mid x_t)\overline{bel}(x_t)$