Using EKP with deterministic and non-deterministic models

[odunbar]

Deterministic setup

For a deterministic forward map G, the current EKP implementation works on the mean \bar{y} and covariance \Gamma of the data. Internally a regularization technique in the form drawing new samples of data for each iteration and ensemble member.

Effectively this solves the problem y = G(\theta) + N(0,\Gamma) by generating samples of \gamma_i ~ N(0,\Gamma) and using samples y_i = \bar{y} + \gamma_i in the algorithm. As we provide \bar{y} we achieve an unbiased result.

Also note - There should be no issue in this formulation working where \bar{y} is replaced by a sample y (and the algorithm still perturbs this internally). In this case one will acheive a (naturally) sample-biased result.

Nondeterministic setup

For nondeterministic forward map G (with mean \bar{G}) we would like to use a sample y and covariance \Gamma here. We do not regularize (redraw the data at every iteration) because the internal variability is sufficient.

Effectively this solves the problem y = G(\theta) = \bar{G}(\theta) + N(0,\Gamma) , where we don't have access to \bar{G}, thus a sample of G already contains the correct variability and no regularization should be performed. As we provide only a sample y this will acheive a (naturally) biased result

To obtain consistency with the current CES framework, the sample of data for EKI and MCMC should be the same.

Solution?

Perhaps the simplest way to do this would be to have a flag deterministic_forward_map=true.

• If true, the sample is perturbed at every iteration and ensemble member (as is traditional)
• If false, the sample is unperturbed.

[ilopezgp]

I believe this was resolved. Should we close @odunbar ?

[odunbar]

Yes we resolved this