this is the algorithm for the case where all the ingredient have these types. In practice, Coming up with an initial $bel$ is a also part of the task.
$N^*$ is supposed to be the normalization of the whole term on the right of it - a normalization to the sum/integral of $bel_{\mathrm in}$. In practice, the latter should normalize to $1$.

$K_u(x,x')$ ought to capture the propagation, possibly determined by actions $u$.

$W_z$ ought to capture a redistribution of believe, due to some observation $z$.

The relation with Bayes rule is discussed in Conditional probability.

Of course may move $W_z$ under the integral too.

This is when the true state evolution is given by a linear relation

$x_k = {F}_k x_{k-1}$.

(+possibly by a noise term)

and when the sensor is set to measure $z_k = {H}_{k} x_k$.

(the H-matrix can be a projection, thus taking into account that you only measure particular features of the truth, and you can't catch em all.)

and you apply the Bayes Filter with

$P_u(x_k, x_{k-1}) = p( x_k \mid x_{k-1}, u_k) = \mathcal{N} ( {F}_k x_{k-1} + B_k u_k, {Q}_k)$

$O_z(x_k) = p( {z}_k\mid x_k) = \mathcal{N}( {H}_{k} x_k, {R}_k)$

where $\mathcal{N}( x, \sigma^2)$ is the normal distribution, except of course with multivariate arguments.

Wikipedia: Gamma function

Function

Factorial function, Conditional probability