A Quasi-Gaussian Approximation for the Probability Distribution of Correlation Functions

Abstract

The likelihood function of correlation functions needs to be known whenever they are used for inference about cosmological parameters. It is usually approximated as a multivariate Gaussian, which is not necessarily a good approximation, as can be seen from the existence of constraints on correlation functions (see (Schneider and Hartlap, A&A 504:705–717, 2009))—thus, a better approximation for the likelihood of correlation functions is required. For a 1-D Gaussian field, the univariate and bivariate likelihood has been derived analytically in (Keitel and Schneider Constrained probability distributions of correlation functions. Accepted for A&A [arXiv:1105.3672] 2011) and can deviate very strongly from Gaussians. Based on the constraints and the exact univariate likelihood, we constructed a quasi-Gaussian ansatz for the multi-variate correlation likelihood which (1) strictly obeys the constraints, (2) yields an approximate Gaussian in cases where the Gaussian approximation for the likelihood holds, and, if this is not the case, (3) provides a much better approximation than the Gaussian, as demonstrated with simulations; finally, (4) it provides a significantly better description than the straightforward copula approach.