| context | $K_u:(X\times X)\to {\mathbb R}$ |
| context | $W_z:X\to {\mathbb R}$ |
| definition | $\Gamma: (X\to {\mathbb R})\to X\to {\mathbb R}$ |
| definition | $bel_{\mathrm out}[bel_{\mathrm in}](x) := N^*W_z(x)\int_A K_u(x,x')\,bel_{\mathrm in}(x'){\mathrm d}x'$ |
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