Include Impose-prior versions of EKP. (a.k.a infinite-time EKI)

[odunbar]

Issue

This results from discussions 4.4 and 4.5 in https://arxiv.org/pdf/2209.11371
The current methodology for EKI / ETKI is built on the formulation of finite-time transport, which provides a nice fast algorithm but with two restrictions
(1) The solution interpretation is only approximately Bayesian only at T=1, (Good if we don't trust priors and so iterate beyond)
(2) The initial ensemble must be sampled from the prior (Bad if prior is very broad / poorly sampled / model is unstable)

The infinite-time variant on the other hand has other properties
(1) It reaches the Bayesian solution at T = infty (good if we do trust priors and want a proper Bayesian solution)
(2) The initial ensemble and the prior are decoupled

Current set-up

The infinite-time variant, can be set-up with modifications from the finite-time setup. (currently can be done in our package via:)
(1) constructing an extended forward map and data such that

• G(u) -> [G(u),u]'
• y -> [y,m]'
• Gamma -> [Gamma 0; 0 C]

addresses the incorrect minimum as the MLE

(2) use covariance inflation with the current covariance matrix: [Check the exact option]
update_ensemble1(... ,additive_inflation = true, additive_inflation_cov = cov(u,dims=2))

addresses collapse resulting in poor convergence

Particularly with (1) this is quite restrictive.

Possible Solution

• The user can optionally add a prior in the Inversion and/or TransformInversion object
• an impose_prior=true flag will be used to extend the state inside the EKI/ETKI update
• investigate whether inflation should be added internally, else is it better to prompt the user to add the correct covariance inflation to ensure quick convergence

Misc

• compute_error! will need to include the prior term
• adaptive scheduler computations may require the prior term
• and possible index issues for update-groups