# Comparing CDFs from a few percentile points using Gaussian Process

I have two distributions I would like to compare. For each distribution I have only x values at some given percentiles (5, 25, 50, 75, 95).

I would like to make some inference on how different these distributions are. I have some idea on how I would like to do it, but may be in a bit over my head technically. My thought is as follows:

1. For each distribution, fit a Gaussian Process from 0 to 1 that would represent the CDF of each distribution conditioning on the known points (I do not know how to restrict the GP so that it would be monotonic, which would be required for a CDF).

2. Obtain a posterior distribution over some difference metric, such as Kolmogorov-Smirnov distance based on "all possible" CDF functions of the distributions given their known values at given percentiles.

I'd like to do it in Stan or PyMc but I would be happy if I could get it done at all. I have only a layman's understanding of GPs so I'm wondering if such an approach is sound.

• GP is not what you want. Since your samples are non-iid Cramér-von Mises test would not work either. As I understand you are trying to use regression to approximate the cdf. As you said GP regression would not give you a monotonic approximation. Look up some parametric forms instead of GP (which is non-parametric). One simple way: Interpolate linearly between the samples so the approximated CDF is guaranteed to be monotonic. It would not integrate to $1$ though. – Seeda Jun 1 '17 at 12:03