# Is the MLE problem for Gaussian Process Regression convex?

Log marginal likelihood for Gaussian Process as per Rasmussen's Gaussian Processes for Machine Learning equation 2.30 is:

$$\log p(y|X) = -\frac{1}{2}y^T(K+\sigma^2_n I)^{-1}y - \frac{1}{2}\log|K+\sigma^2_n I|-\frac{n}{2}\log2\pi$$

Implementations typically apply gradient descent by default to maximize this over some family of $K$, where that is typically a sum/product of constant kernels, RBF kernels, periodic kernels, etc

It is not clear how convex this optimization problem is, in general. If we sample some data $X$, $y$ from a 'nice' process (i.e. preventing crazily-contrived examples of non-convexity that would never occur in real life data), is this problem convex for common kernels? (sums/products of constant, RBF, exponential sine-squared, etc)

• To get started, see "Covariance estimation for Gaussian variables" beginning on p. 155 and exercise 7.4 in "Convex Optimization",Boyd and Vandenberghe, web.stanford.edu/~boyd/cvxbook – Mark L. Stone Apr 23 '18 at 23:02