Hi!

I recently tried to play with nested sampling in Radvel and so far it's working pretty well. If this gets merged eventually, it would fix #149.

Things already done

The PR includes the following:

• Prior transform (inverse CDF) method implemented for most prior distributions.
• Prior transform method implemented for the Posterior class. It iterates over free parameters and applies the inverse CDF of their prior to the unit cube.
• A check_proper_priors() method for Posterior, performing some checks to make sure there are no "improper" priors, and that each parameter has only one prior.
• A draft example copying the K2-24 one but with nested sampling (dynesty) instead of MCMC. This is just a temporary example I used for development. Another one could be used to make tutorials more diverse.

Things left to do

Some things I was wondering and/or that are still left to do:

• Should there be a nested sampling module similar to the MCMC one that wraps the sampler and provides utility function? If so should it aim to support a single sampler among dynesty, Ultranest, etc., or more than one? Another option would be to simply provide prior transforms and let users run the sampler themselves. The main upside of a dedicated module is probably integration in the CLI interface.
• How should joint/non-finite/multiple priors be handled? See note below.
• Some priors still don't have a transform implemented, either because they were tricky (see my last point) or because I had never used them and did not look into them yet for this quick draft PR.
• The check_proper_priors() method is never run automatically for now. It could be run on every prior_transform call, integrated into a nested_sampling.py module, or documented clearly in the tutorial to let users run it just once before running the sampler.
• There is no nested sampling/prior documentation yet.
• I still need to write a proper nested sampling tutorial

Question(s) about priors

One thing I found tricky is how to handle certain priors that don't directly depend on one parameter or that complement other priors (e.g. ecc < 1, K > 0).
Some avenues I thought about to handle those are:

• Using truncated distributions when the parameter is explicitly defined anyways (e.g. instead of no prior on K and K > 0, use ModifiedJeffeys, HardBounds, a log parametrization, etc.)
• Using a "2D" prior when the parameter is defined as a function of others. For example, replace uniform priors on se{cos}w + EccentricityPrior by a "unit disk" prior (see e.g. the exoplanet package)
• A generic way to handle multiple priors on the same parameters or on dependent parameters. I see two ways of doing this:
  • Incorporating the "extra" priors in the likelihood so that they return -inf when the parameter is out of bounds (this is what juliet does with the modelOK attribute I think)
  • A generic way to combine priors when building the model. I'm not sure how this would work for inverse CDFs yet though.

I'm open to any comments/suggestions about all of the above of course.

Thank you!