# Is the gaussian process non-monotone

The Gaussisna process (GP) is one of the most famous non-parametric regression techniques. The gaussians process is defined as a collection of random variablers with a joint multivariate gaussian distribution.

I have read that the gaussian process is non-monotone. Is that always true, i.e. is the GP non-monotone with probability 1 ?

Is there a way to show that the Gaussian process can be monotone is some cases i.e (for any $x\leq y$ we have that $f(x)\leq f(y)$. Or can this be disproven ?

• For the benefit of the readers, you should say exactly what you mean by non-monotone, in the context of a Gaussian Process. Sep 18, 2016 at 17:12
– Wis
Sep 18, 2016 at 19:04
• Are x and y vectors of dimension $> 1$ (in general)? If so, what does $x \le y$ mean? Sep 18, 2016 at 19:13
• @MarkL.Stone. They are just scalar values, I did not add a bold font for that reason. Thanks
– Wis
Sep 18, 2016 at 19:33
• The mean function of a Gaussian process (defined on $\mathbb{R}$) can be monotone: Just add a monotone function to a zero mean process. But since $f(x)$ and $f(y)$ are bivariate normal, the probability $P(f(x)>f(y))$ will be always larger than zero, no matter what covariance function you choose (except for degenerate cases such as correlation constant -1 or similar)
– g g
Sep 18, 2016 at 22:19

Yes, a Gaussian Process (GP) is strictly non-monotonic. This is easy to show for the case that the kernel is non degenerate (ie full rank) as the posterior of any unobserved point will have a posterior distribution which is normally distributed between hence spans $[-\infty, \infty]$. As such the probability of $y < \hat{y}$ for any given $\hat{y}$ is strictly greater than zero.