Two papers written by astronomers have been recently posted on arXiv about (new) ways to approximate evidence. Since they both perceive those approximations as some advanced form of quadrature, they are close enough that a comparison makes sense.
The paper by Rutger van Haasteren uses a Voronoi tessellation to represent the evidence as
when the 
where 
The second paper by Martin Weinberg also adopt a quadrature perspective, while integrating in the Lebesgue sense rather than in the Riemann sense. This perspective applies to nested sampling even though John Skilling does not justify nested sampling that way but Martin Weinberg also shows that the (infamous) harmonic mean estimator also is a Lebesgue-quadrature approximation. The solution proposed in the paper is a different kind of truncation on the functional values, that relates more to nested sampling and on which I hope to report more thoroughly later.
