**I**n the 19 Jan 2017 issue of Nature [that I received two weeks later], a paper by Tarnita et al discusses regular vegetation patterns like fairy patterns. While this would seem like an ideal setting for point process modelling, the article does not seem to get into that direction, debating instead between ecological models. Which combines vegetal self-organisation, with subterranean insect competition. Since the paper seems to derive validation of a model by simulation means without producing a single equation, I went and checked the supplementary material attached to this paper. What I gathered from this material is that the system of differential equations used to build this model seems to be extrapolated by seeking parameter values consistent with what is known” rather than estimated as in a statistical model. Given the extreme complexity of the resulting five page model, I am surprised at the low level of validation of the construct, with no visible proof of stationarity of the (stochastic) model thus constructed, and no model assessment in a statistical sense. Of course, a major disclaimer applies: (a) this area does not even border my domains of (relative) expertise and (b) I have not spent much time perusing over the published paper and the attached supplementary material. *(Note: This issue of Nature also contains a fascinating review paper by Nielsen et al. on a detailed scenario of human evolutionary history, based on the sequencing of genomes of extinct hominids.)*

## Archive for Voronoi tesselation

## lords of the rings

Posted in Books, pictures, Statistics, University life with tags ants, fairy rings, Namibia, Nature, point processes, termites, Voronoi tesselation on February 9, 2017 by xi'an## Quadrature methods for evidence approximation

Posted in Statistics with tags Bayesian model choice, evidence, harmonic mean estimator, Lebesgue integration, nested sampling, Voronoi tesselation on November 13, 2009 by xi'an**T**wo 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.

**T**he paper by Rutger van Haasteren uses a Voronoi tessellation to represent the evidence as

when the ‘s are simulated from the normalised version of and the ‘s are the associated Voronoi cells. This approximation converges (even when the ‘s are not simulated from the right distribution) but it cannot be used in practice because of the cost of the Voronoi tessellation. Instead, Rutger van Haasteren suggests using a sort of an approximate HPD region and its volume, , along with an harmonic mean within the HPD region:

where is the total number of simulations. So in the end this solution is actually the one proposed in our paper with Darren Wraith, as described in this earlier post! It is thus nice to see an application of this idea in a realistic situation, with performances that compare with nested sampling in its MultiNest version of Feroz, Hobson and Bridges. (This is especially valuable when considering that nested sampling is often presented as the only solution to approximating evidence.)

**T**he 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.