record hyperparams selected within draw-gpr

[reedessick]

within each drawn EOS, we want to record the hyperparameters associated with that draw. In this way, assuming we weigh samples from various drawn EOS, we can generate both prior and posterior distributions on the EOS hyperparameters, which could be a quick way to infer length scales without cumbersome model selection based on "by-hand" divisions of hyper-parameters.

[reedessick]

Note, if we obtain posterior processes on the hyperparameters, these can naturally be used to set define updated priors when drawing new EOS in a stacking experiment. This could be computationally useful to avoid the acceptance fraction of our Monte Carlo integration vanishing as the EOS drawn from an initially wide prior become more and more constrained as more observations are made.

Would need to test whether this behaves as expected, but could be a relatively "clean" path forward

[reedessick]

In the case of many observations, we may get many constraints at relatively low densities (and favor small length scales there) but few observations at high densities (and favor long length scales there), or start to see phase transitions, etc which require an effectively different length scale at different densities. We could then adopt a local-expert model to allow us to specify this in a somewhat tractible way, but that will probably cause the number of hyperparameters explode in a bad way...

[reedessick]

this functionality was added in 88890da9a069be9b8142d3075b678242459ac865. However, we do not currently have an automated way to extract these. Let's cross that bridge when it comes, though.