First of all hello. I am trying to understand the kalman Filter using Bayesian probabilities. For example let's say that we have
$$P(Xt/Yt, \text{old data})= \frac{P(Yt\mid Xt) P(Xt\mid\text{old data})}{P(Yt\mid \text{old data})}$$ or $$ P(Xt/Yt,\text{old data})= n P(Yt\mid Xt) P(Xt\mid \text{old data})$$
where $n$ is for normalization. So let's say that someone gives us $Y=$"red" observation then this becomes
$$P(Xt/\text{red}, \text{old data}) = n P(\text{red}\mid Xt) P(Xt\mid \text{old data})$$
So we multiply the already known state estimation with the function P(Y|X) with $y$ held constant (after all we have observed $y$). So let's say in the kalman filter we already have a gaussian estimation of $Xt$, let's say after some action $u$. This is $P(Xt|\text{oldData})$ where $\text{oldData}=(\text{olderData},u)$.
Then logically we have to multiply it by a gaussian estimation of $P(y|X)$ probability and then normalize it. But the problem is I don't know how the kalman filter does this. In the kalman filter we multiply with a gaussian $~N(y;CXt,Σ)$.
I can't understand how this is equal to $P(y|X)$ since it seems to me that the gaussian is plotted on $y$ and $Xt$ is known. So the gaussian is equal to something like this
$Y=aX+b$
This would make sense if we would like to find the $P(Y|x)$ with $x$ held constant but we need $P(y|X)$ because our $y$ is known from the sensors already. I hope that makes sense lol.