# "interpolating" a distribution between two existing distributions that are spatially correlated

Hello pyMC community. My first post here, I just started using pyMC and very excited to do some bayesian inference with it. My question is a bit more general on statistics, and I could not find it already answered here, or the proper tool to use.

Problem background:
I am building prior distributions using a forward model. The model computes a plane that may have curvature, or may deviate slightly in its orientation (these parameters are randomly sampled and based on “realistic” estimates). I then look at how this plane intersects a line, and how my prediction if this intersection may be off based on the prior uncertainty (plane curvature or orientation uncertainty that I can not know). So essentially I am sampling the prior distribution.

Sampling strategy
The line can be at any distance from the plane’s origin and I can not sample all the possible distances. But the distributions should behave well between the sampled points. My questions is, what is the proper way to interpolate a distribution between two known distributions? Say I have an estimate of the prior uncertainty at 30 m from the plane origin and one at 40 m. These have been sampled using several realizations of possible planes. Now I want to get the uncertainty at 35 m. What’s a good way to do this?

Attached is a violin plot of how these distributions look as a function of distance, just to get a feel. The dotted line in the middle is the intersection of a horizontal plane with no curvature. Also a screenshot of how this geometry actually looks, with one plane and two curved surfaces. Thank you for your help and time!

Hello, just to inform that I have found a solution for this here using the python optimal transport package. This seems to work well with wasserstein barycenter interpolation. Not a pure pyMC solution but now I can go into MCMC sampling with these distributions using pyMC!