Archive for Planck Collaboration

nested sampling: any prior anytime?!

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , , , , , on March 26, 2021 by xi'an

A recent arXival by Justin Alsing and Will Handley on “nested sampling with any prior you like” caught my attention. If only because I was under the impression that some priors would not agree with nested sampling. Especially those putting positive weight on some fixed levels of the likelihood function, as well as improper priors.

“…nested sampling has largely only been practical for a somewhat restrictive class of priors, which have a readily available representation as a transform from the unit hyper-cube.”

Reading from the paper, it seems that the whole point is to demonstrate that “any proper prior may be transformed onto the unit hypercube via a bijective transformation.” Which seems rather straightforward if the transform is not otherwise constrained: use a logit transform in every direction. The paper gets instead into the rather fashionable direction of normalising flows as density representations. (Which suddenly reminded me of the PhD dissertation of Rob Cornish at Oxford, which I examined last year. Even though nested was not used there in the same understanding.) The purpose appearing later (in the paper) or in fine to express a random variable simulated from the prior as the (generative) transform of a Uniform variate, f(U). Resuscitating the simulation from an arbitrary distribution from first principles.

“One particularly common scenario where this arises is when one wants to use the (sampled) posterior from one experiment as the prior for another”

But I remained uncertain at the requirement for this representation in implementing nested sampling as I do not see how it helps in bypassing the hurdles of simulating from the prior constrained by increasing levels of the likelihood function. It would be helpful to construct normalising flows adapted to the truncated priors but I did not see anything related to this version in the paper.

The cosmological application therein deals with the incorporation of recent measurements in the study of the ΛCDM cosmological model, that is, more recent that the CMB Planck dataset we played with 15 years ago. (Time flies, even if an expanding Universe!) Namely, the Baryon Oscillation Spectroscopic Survey and the SH0ES collaboration.