The quick answer is no, don’t read too much into optimums. Here’s more if you want: Typical Sets and the Curse of Dimensionality
Have you looked at posterior intervals for rho, alpha, and sigma? I’d only be alarmed by the variation if the uncertainty in those variables was really really small (and consistently did not include the parameters you used to generate the data with). Also, an Rhat like 1.029 is suspicious. You want these to be really close to 1.0 (1.01 is proposed as a rule-of-thumb in the new Rhat stuff: Maybe it’s time to let the old ways die; or We broke R-hat so now we have to fix it. | Statistical Modeling, Causal Inference, and Social Science)
Is G the number of data points (N)? Or to be clearer, is the number of points in the Gaussian process scaling with the number of observations?
If so the Cholesky decomposition is going to be a problem. That is expensive (assuming the matrix is GxG, it scales like G^3).
The as N → infinity sorta stuff usually assumes fixed numbers of parameters. If the number of parameters scales with the data, then you probably don’t have any guarantees.
This sounds a lot like something you wanna write out as some sort of regression model, especially considering you have a ton of data. Linear models n’ such are the things that are gonna scale. Have you seen this book: http://www.stat.columbia.edu/~gelman/arm/ ? It’s a lot of creative regression stuff. The models are available as Stan models too: ARM Models · stan-dev/example-models Wiki · GitHub .