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I was reading the notes on Gaussian Processes by Choung B. Do (stanford course CS229) however was unsure of how the mean function worked and what a random variable was on the Gaussian Process

So first it reminds us of what a stochastic process is:

enter image description here

Which I think makes sense to me. $x \in \mathcal{X}$ is an index set and we are indexing the random variables. So for example, if $\mathcal{X} = R$, then, $f(0)$ would be the random variable that takes some random value (maybe $f(0) = f = 6$) at time step 0. Similarly, $f(2.3)$ would be a random variable (with a distribution) at time step 2.3. I think that makes sense.

Then it goes on to define a Gaussian Process more carefully:

enter image description here

I think that also makes sense, as that is just saying how each of the random variables depend with respect to each other and $m(x_1)$, I suppose, is just the mean wrt to time step $x_1$.

Then it makes the remark that confuses me:

enter image description here

This is exactly the point that confuses me because, it says that the mean $m(x) = E[x]$. In my head, $x$ was just an element in the index set and therefore it is not random (intuitively, if we are taking samples from a stochastic processes, we get samples at a finite set of time intervals that we decide, x is not usually considered random). It is a constant. What is random is $f(x)$. So for me what would make sense is to say:

$$m(x_i) = E[ f(x_i) ]$$

that seems to be the correct statement to me. Am I wrong or is there something crucial I am misunderstanding since this discussion should be pretty basic?

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1 Answer 1

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Your understanding is correct. There is apparently mistake in the notes and the equations should be \begin{array} mm (x) &= E[ f(x) ], \\ k(x,x') &= E[(f(x)-m(x))(f(x')-m(x'))]. \end{array}

For reference, see Equation (2.13) in page 13 of C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006, available online.

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  • $\begingroup$ I was about to point out the typo :P $\endgroup$ Commented May 6, 2015 at 15:14
  • $\begingroup$ this typo take a half hour from me!!thanks for OP and answerer. $\endgroup$ Commented Aug 13, 2015 at 17:38

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