# Gaussian Processes for convex functions

As I understand, when using Gaussian Processes (GP) for regression, one can/should incorporate prior knowledge about the function into the GP. Let's say I have a nonlinear function $f: \mathbb{R}^n \to \mathbb{R}$ I want to learn from samples $(x_i, y_i = f(x_i) + \epsilon)$ using GP. I also know that $f$ is a convex function. Is there any way to incorporate this information into the GP or the tuning of the GP's hyperparameters?

For example, I may want to enforce that the resulting GP from tuning the hyperparameters is "convex" in the following sense:

• The mean prediction as a function of $x$ is convex; and/or
• The posterior distribution of functions of $x$ (the function-space view as discussed in the GPML book) gives low probability to non-convex functions. I don't yet have a formal mathematical definition of this criterion.

Any ideas or references are appreciated. Or if you think this question is better posted to another forum, please let me know.

Assume that $n=1$ and that your GP has twice-differentiable samplepaths.