On the Edgeworth Series…

There’s a nice paper on the arXiv today by Elena Sellentin, Andrew Jaffe and Alan Heavens about the use of the Edgeworth series in statistical cosmology; it is evidently the first in a series about the Edgeworth series.

Here is the abstract:

Non-linear gravitational collapse introduces non-Gaussian statistics into the matter fields of the late Universe. As the large-scale structure is the target of current and future observational campaigns, one would ideally like to have the full probability density function of these non-Gaussian fields. The only viable way we see to achieve this analytically, at least approximately and in the near future, is via the Edgeworth expansion. We hence rederive this expansion for Fourier modes of non-Gaussian fields and then continue by putting it into a wider statistical context than previously done. We show that in its original form, the Edgeworth expansion only works if the non-Gaussian signal is averaged away. This is counterproductive, since we target the parameter-dependent non-Gaussianities as a signal of interest. We hence alter the analysis at the decisive step and now provide a roadmap towards a controlled and unadulterated analysis of non-Gaussianities in structure formation (with the Edgeworth expansion). Our central result is that, although the Edgeworth expansion has pathological properties, these can be predicted and avoided in a careful manner. We also show that, despite the non-Gaussianity coupling all modes, the Edgeworth series may be applied to any desired subset of modes, since this is equivalent (to the level of the approximation) to marginalising over the exlcuded modes. In this first paper of a series, we restrict ourselves to the sampling properties of the Edgeworth expansion, i.e.~how faithfully it reproduces the distribution of non-Gaussian data. A follow-up paper will detail its Bayesian use, when parameters are to be inferred.

The Edgeworth series – a method of approximating a probability distribution in terms of a series determined by its cumulants – has found a number of cosmological applications over the years, but it does suffer from a number of issues, one of the most important being that it is not guaranteed to be a proper probability distribution, in that the resulting probabilities can be negative…

I’ve been thinking about how to avoid this issue myself, and mentioned a possibility in the talk I gave at South Kensington Technical Imperial College earlier this summer. The idea is to represent the cosmological density field (usually denoted δ) in terms of the square of the modulus of a (complex) wave function ψ i.e. |ψψ*|. It then turns out that the evolution equations for cosmic fluid can be rewritten as a kind of Schrodinger equation. One powerful advantage of this approach is that whatever you do in terms of approximating ψ, the resulting density ψψ* is bound to be positive. This finesses the problem of negative probabilities but at the price of introducing more complexity (geddit?) into the fluid equations. On the other hand, it does mean that even first-order perturbative evolution of ψ guarantees a sensible probability distribution whereas first-order evolution of δ does not and has



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