===== Bayes algorithm =====
==== Function ====
| @#55CCEE: context     | @#55CCEE: $K_u:(X\times X)\to {\mathbb R}$ |
| @#55CCEE: context     | @#55CCEE: $W_z:X\to {\mathbb R}$ |
| @#FF9944: definition  | @#FF9944: $\Gamma: (X\to {\mathbb R})\to X\to {\mathbb R}$ |
| @#FF9944: definition  | @#FF9944: $bel_{\mathrm out}[bel_{\mathrm in}](x) := N^*W_z(x)\int_A K_u(x,x')\,bel_{\mathrm in}(x'){\mathrm d}x'$ |

>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 $1$.

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=== Discussion ===
$K_u(x,x')$ 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 ]].

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


== Kalman filter ==
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.

=== Theorems ===


=== Reference ===
Wikipedia: [[https://en.wikipedia.org/wiki/Naive_Bayes_spam_filtering |Gamma function]]

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=== Context ===
[[Function]]
=== Related ===
[[Factorial function]],
[[Conditional probability ]]