Gaussian process and Gaussian regression


Hello, I have some problem understing the Gaussian regression and the way it works.

Could you please explain to me some things?

The first thing I don't understand is why this regression does it use the joint/multivariate probability to get new value of the Y?

The pic:

The definition made by wikipedia
Given random variables X, Y, ..., that are defined on a probability space, the joint probability distribution for X, Y, ... is a probability distribution that gives the probability that each of X, Y, ... falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution.

What does this have to do with the regression, where the Y is supposed to be the dependent variable we want to guess and the x the predictors?

Why the gaussian regression does it joints the Y and X toghter to find the conditional mean? Could you please explaine? Thanks


No cake for spunky
By gaussian do you mean linear regression?

I believe that the joint probability reflects how you solve for Y given a set of X. Most practitioners ignore the theory behind the regression which is what this deals with.