A quasi-Gaussian approximation for the probability distribution of correlation functions

TL;DR

This paper introduces a quasi-Gaussian approximation for the probability distribution of correlation functions, improving over the Gaussian assumption and potentially enhancing cosmological parameter inference.

Contribution

It develops a transformation-based quasi-Gaussian approximation that better matches the true distribution of correlation functions compared to previous methods.

Findings

Quasi-Gaussian PDF closely matches simulation results.

It outperforms the copula approach in accuracy.

Different results in Bayesian analysis suggest impact on cosmological inference.

Abstract

Context. Whenever correlation functions are used for inference about cosmological parameters in the context of a Bayesian analysis, the likelihood function of correlation functions needs to be known. Usually, it is approximated as a multivariate Gaussian, though this is not necessarily a good approximation. Aims. We show how to calculate a better approximation for the probability distribution of correlation functions, which we call "quasi-Gaussian". Methods. Using the exact univariate PDF as well as constraints on correlation functions previously derived, we transform the correlation functions to an unconstrained variable for which the Gaussian approximation is well justified. From this Gaussian in the transformed space, we obtain the quasi-Gaussian PDF. The two approximations for the probability distributions are compared to the "true" distribution as obtained from simulations. Additionally, we test how the new approximation performs when used as likelihood in a toy-model Bayesian analysis. Results. The quasi-Gaussian PDF agrees very well with the PDF obtained from simulations; in particular, it provides a significantly better description than a straightforward copula approach. In a simple toy-model likelihood analysis, it yields noticeably different results than the Gaussian likelihood, indicating its possible impact on cosmological parameter estimation.