I would like to compare a parametric function to a Gaussian Process. This may sound weird, but read on:
Data description. I am looking at projections of a 3D object. However I expect a certain amount of smoothness in the true object, so it seems natural to use a GP to represent it, thinking of each image pixel as a (partial) observation of every voxel in the final object. So I've fitted my GP.
Now I want to do inference on that GP. Why? Because even though I have inferred the shape, I need to know what it actually is made of (FYI, I'm inferring a molecule's components from the observed outline). This means I need to fit a parametric function (actually a Gaussian Mixture Model) to a GP. So I would like to write some kind of likelihood of the GP given a GMM.
I'm open to suggestions, including using something besides a GP to represent my data. But note that it is impractical to fit the final model directly to raw data, because that model has many additional priors so it takes a long time to sample. The intermediate data processing is a kind of data reduction.