Understanding a paper on Gaussian Processes

Question

I am reading and trying to understand the paper "Gaussian Processes for Signal Strength-Based Location Estimation" by Ferris et al. (A copy of the paper may be accessed here). At the moment I am stuck with Section II A. All notation used below is consistent with the paper.

1. The authors introduce two covariance functions in equations 3 and 4. Can anyone explain the difference between these two functions? (i.e. why they are useful.)

2. In equations 5 and 6, the authors use $\textbf{k}_*$, which is "the $n\times1$ vector of covariances between $x_*$ and the training inputs $\textbf{X}$," I don't understand how to compute it though. Any insights? (Is it found from equation 4?)

I'm not very familiar with the theory of Gaussian Processes, and unfortunately don't have the luxury of time to spend understanding the theory. I would appreciate answers that focus on the "functional" aspects of the paper, as that is the reason I am reading it.

Answer 1

These two lectures by Nando de Freitas follow the same notation and will give you a full understanding of Gaussian Processes: Introduction to GP, Active learning with GP.

Answer 2

1. Eq.3 is in scalar notation, Eq.4 in matrix notation; they are practically equivalent. You use the second one when you code up things usually when trying to vectorize your implementation; also it is easier to write down on the board. :)
2. You compute it using Eq. 3 (or .4); it is the covariance of your point of estimation $x_*$ and the known/input put in $X$. So practically you compute Eq.3 iterating over all $x_i$ (something like $cov(y_p,y_*) = k(x_p,x_*)$ and you end up with a vector $k_*$.

Without wanting to insult you, your questions are basic to the point it seems you don't understand standard notation. Consider spending a couple of hours on GP at least by familiarizing yourself with the notation used. It will be time extremely well-spend. The lectures suggested by @juliohm can be a decent starting point. This short tutorial by Eden : Gaussian Processes for Regression: A Quick Introduction, is the simplest and most direct material I can think of for getting out quickly somewhat up to speed with GPs, it should take you less that an hour or two I think. Good luck and don't hesitate to ask more things!