[Theoretical] Could nested sampling techniques exploit multilevel / multifidelity models?

[jbrezmorf]

I'm dealing with inversion for costly PDE models. I'm able to create different discretizations (multilevel model) and/or
train some surrogate that is even less precise.
Our team has some experience with Delayed Acceptance MCMC techniques that can utilize cheaper approximations.
After some unsuccessful experiments with parallel MCMC samplers, we want to try nested sampling, which seems better suited for parallelization.

Do you know about nested sampling techniques that could use multi-fidelity models?

I naively consider sampling using the surrogate and running the full model only if the sample with likelihood improvement is found to confirm it. That would be similar to DA MCMC. But I'm completely new to nested sampling and can not judge if such an approach is viable.

[JohannesBuchner]

Perhaps the least complex approach is to run nested sampling with the surrogate model, and then take the posterior samples and importance sample them (reweighing by L_full/L_surrogate). You can also re-estimate the evidence this way. You can use the weighted_post.txt output file.

[JohannesBuchner]

A more refined approach integrated to nested sampling is to implement a likelihood function that

1. evaluates Lup, an upper limit to the likelihood(theta), using the surrogate model
2. checks if this exceeds the current sampler log-likelihood threshold (accessible from sampler.Lmin)
3. if not, returns an arbitrary value smaller than sampler.Lmin, such as Lup -> the point will be discarded by nested sampling.
4. if yes, calls the full model to get a refined log-likelihood and returns it.

You can do this by implementing a class with __call__ for the log-likelihood, pass it to the sampler, and between instantiating the sampler and calling sampler.run pass the class the sampler object, so that it can in the loglikelihood call query the current Lmin.