Run MCMC in the reduced input space

[odunbar]

Issue

Currently - as the primary bottleneck, the dimension reduction creates a low dimensional map f from pairs (x, G(x)) such that

G(x) = D o f o E (x), for encoder-decoder pairing E,D that possibly reduce dimensionality.

When predicting at a new x' the current Emulator.predict() method performs f o E(x') with optional decoding.

During sampling for MCMC we typically use this predict method.

For full efficiency in sampling too, we should really be projecting x |-> z = E(x) and sampling over the encoded z, then projecting back to x at the end.

Solution

• Possibly remove the encoder_schedule from the emulator storage - have it initialized there and passed back
• Pass the encoder_schedule to the `MCMCWrapper
• Encode not only the observation in MCMC but also the prior and IC.
• sample as normal with the reduced prior, but when returning the posterior from the MCMC, sample over the prior to project back into the original space