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A paper that I am reading mentions the likelihood for a squared-exponential-kernel Gaussian Process with no data. In particular, when the kernel function is:

$$ F(x_i, x_{i'}) = v_0\exp\Big(-\frac{(x_i - x_{i'})^2}{2l^2}\Big) + v_1\delta(i, i') $$

then the likelihood is a Gaussian with zero mean and variance $v_0 + v_1$, where $v_0$ is the amplitude and $v_1$ is the white noise variance.

I'm curious as to how this is derived. How would one calculate the probability that a data point belongs to a particular Gaussian Process, if it has no data?

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Before we collect any data, the prior distribution in a GP is Gaussian with mean 0 and covariance $F(x_i, x_{i'})$, to follow your notation.

For any single point, its variance under the prior is simply $v_0 + v_1$.

For more detail, you can look at Chapter 2 of Rasmussen's Gaussian Processes for Machine Learning.

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  • $\begingroup$ I see, I understand why the variance under the prior is $v_0 + v_1$. However, why are we using the variance under the prior as the value for the likelihood? In particular, why the variance? $\endgroup$ – peco May 30 '17 at 14:23
  • $\begingroup$ You're using the variance under the prior because you haven't collected any data yet! $\endgroup$ – Kevin Yang Jul 3 '17 at 21:01
  • $\begingroup$ Thanks for the reply! It's a misconception on my part: we are not using $v_0 + v_1$ as the expression of the likelihood. The actual expression for the likelihood is the Gaussian pdf $N(0, v_0 + v_1)$! $\endgroup$ – peco Jul 4 '17 at 10:38

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