Bayesian Filtering

Predict (total probability over the motion model):

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

Update (Bayes' rule with the measurement model):

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

Every filter — KF, EKF, histogram, particle — is this same loop with a different representation of $bel$.

The recursion only works because the state is assumed complete: $x_t$ depends on the past only through $x_{t-1}$ and $u_t$, and $z_t$ depends on the past only through $x_t$.

This conditional independence is what lets you replace a growing history of data with a single running belief.

Predicting through the motion model always adds uncertainty (it convolves in process noise), even with a perfect model.

Updating with a measurement always removes uncertainty, weighted by how much you trust the sensor. A filter that only predicts drifts; one that only updates never smooths out noise.

The Bayes filter itself is intractable to compute exactly for general distributions — every concrete filter is a choice of how to represent and propagate $bel(x_t)$: a single Gaussian (KF), a linearized or sigma-point Gaussian (EKF/UKF), a discretized grid (histogram filter), or a weighted sample set (particle filter).

Gaussian + linear motion/measurement models collapse the integral into closed form — that's the Kalman filter.

Drop linearity but keep the belief unimodal and you get the EKF/UKF. Drop the Gaussian assumption entirely and you need a grid or particles to represent arbitrary, possibly multimodal shapes.