Gaussian Process conditioning $X|Y \sim N(\mu_{X}+\sum_{XY}\sum_{YY}^{-1}(Y-\mu_Y), \sum_{XX}-\sum_{YY}^{-1}\sum_{YX})$ I found this formula in a post about Gaussian Proccess: https://distill.pub/2019/visual-exploration-gaussian-processes/
$X|Y \sim N(\mu_{X}+\sum_{XY}\sum_{YY}^{-1}(Y-\mu_Y), \sum_{XX}-\sum_{YY}^{-1}\sum_{YX})$
This is kind of weird notation cause of how Latex formats it but $\sum_{MM}, M\in{X,Y}$ is just the covariance between the variables cause $\sum$ represents the covariance matrix and $MM$ is just the index. The $\sum^{-1}$ just means the inverse. So the above eqaution is state that $X$ conditoned on $Y$ the probability on one depending on the other is a Gaussian with mean=$\mu_{X}+\sum_{XY}\sum_{YY}^{-1}(Y-\mu_Y)$ and standard deviation equal to: $\sum_{XX}-\sum_{YY}^{-1}\sum_{YX}$. This is applied to a 2D multivariate Gaussian distribution with variables X and Y.
I don't quite understand why this is. For the mean, the covariance between XY divided by the variance of Y, multiplied by Y - the mean of Y? I'm not entirely sure what Y - the mean of Y means here. I don't know if it's all the points - the mean? I don't entirely understand how sampling from this equation gives $X|Y$
 A: you can find the same formula in Wikipedia, they explain this in some details
https://en.wikipedia.org/wiki/Multivariate_normal_distribution
I will give you some intuition. First of all, this is a multivariate notation, that's why you use a matrix notation. This equation shows a conditional distribution of $X$ given $Y$. This means that without knowing the $Y$ our $X$ just follows $\mathcal{N}(\mu_x, \Sigma_x)$, however, if you know the $Y$ you can update the distribution of $X$. Imagine that $X$ - salary and $Y$ - age. Knowing that $Y = 30$ gives your completely different expected value for the salary relative to observing $Y = 1$.
This is the reason, why you adjust the mean of $X$ by $\Sigma_{XY}\Sigma_{YY}^{-1}(Y - \mu_Y)$, the first term just measures the covariance between $X$ and $Y$. If there is no covariance you will have no adjustment. The second term adjusts the covariance, because $Y$ could be very volatile, but barely related to $X$. The third term just shows whether we deviate above or below the mean of $Y$ because our reference point is $\mu_Y$ if we do not know $Y$.
