# Likelihood for a Gaussian Process with no data

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?

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.

• 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?
– peco
Commented May 30, 2017 at 14:23
• You're using the variance under the prior because you haven't collected any data yet! Commented Jul 3, 2017 at 21:01
• 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 $N(0, v_0 + v_1)$!
– peco
Commented Jul 4, 2017 at 10:38