# Gaussian Process - How to interpret the posterior?

Let the prior on the regression function $$f(·)$$ be a GP, denoted by $$f(x) ∼ GP (m(x), κ(x, x'))$$. $$m(x)$$ is usually $$0$$, so let us consider that too.

From what I understand a Gaussian Process can be seen as a prior over a set of functions.

I have a set of observations $$\{(x_1,y_1), (x_2,y_2), ..., (x_N,y_N)\}$$ and I want to predict $$y^*$$ for my specific $$x^*$$. I constrain the GP on my observations and then I predict $$y^* = f(x^*)$$.

My question is what happens to my mean and covariance $$m(x), κ(x, x')$$? I initially gave them a value, but did that change when I constrained the GP? In another words, is $$m(x), κ(x, x')$$ different from the values I set initially?

• So can I get a formula dependent on the original $m(x)$ and on my data to get the new posterior mean? I understand that this is also a function. – Catarina Alves Feb 19 at 21:40
• In a regression with a Gaussian likelihood your posterior mean is $E[f^*|X,y,X^*] = K(X^*,X)[K(X,X) + \sigma^2 \mathbf(I)]^-1(\mathbf{y} - m(X)) + m(X^*)$. So the only elements that change if you want to predict at new data points is the cross kernel function and the mean function for the test data points. Think of that unchanged term as like the real computation of a GP regression. So yeah you do get a formula that is dependent your data as well as the functions that you defined for your original training data set. – Chango Feb 20 at 8:49