Is $K(s,t) = \log(2 + |s - t|)$ a kernel for a Gaussian process?

Question

Is there a Gaussian process with a kernel $E[X_s X_t] = K(s, t) = \log(2 + |s - t|)$? It's symmetric, it's non-negative, and yet it seems like it shouldn't be.

Answer 1

I figured this out on my own. The answer is no.

This kernel is not semi-positive definite. Take $s = t + h$ with $h > 0$; then consider the sum:

$$\sum_{i, j \in \{t, t + h\}} c_i c_j K(i,j)$$

where $c_i = 1$ if $i = t + h$ and $c_i = -1$ if $i = t$. This form is associated with $Var(X_{t + h} - X_t)$. It can be rewritten as:

$$K(t + h, t + h) - 2K(t+h, t) + K(t,t) = 2\log(2) - 2 \log(2 + h) = 2\log\left(\frac{2}{2 + h} \right)$$

This is negative for any $h > 0$ and so $K$ is not semi-positive definite, which is required in order for a function to be the kernel of a Gaussian process.

One interpretation of the semi-positive definite requirement is that it should not be possible to compute a negative variance for a linear combination of $X_t$ in the Gaussian process. A kernel must satisfy this, and this kernel does not; negative variances would be possible.