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

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 of one-dimensional random fields, which we call "quasi-Gaussian". Methods. Using the exact univariate probability distribution function (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.

Cosmology with cosmic shear observations: a review

“…In contrast, analytical approaches might be promising to determine the true likelihood function. Transforming the data to obtain more Gaussian distributions involves Gaussianizing the convergence (Joachimi et al 2011b, Seo et al 2011, Yu et al 2012), or transforming the correlation function , Wilking & Schneider 2013. The so-called copula can be used to reconstruct the multi-variate probability distribution function (pdf) from one-dimensional pdfs (Sato et al 2011a).…”

“…Reconstructing the projected mass (or related quantities such as the density or the lensing potential) in large, blind fields, where the shear is an order of magnitude smaller than for galaxy clusters, is more challenging. Wilson et al (2001) constructed κ maps on UH8K/CFHT data, which were correlated with galaxy light to infer mass-to-light ratios on large scales. Convergence maps from one of the DLS fields (Wittman et al 2006) were confronted in Geller et al (2005) with a velocity dispersion map from 10, 000 galaxy spectra obtained with Hectospec/MMT (Magnum Mirror Telescope).…”

Likelihood of the Power Spectrum in Cosmological Parameter Estimation

“…Kilbinger et al 2013) and a mild effect on baryon acoustic oscillations (BAO) analyses (Labatie et al 2012). Separately, Wilking & Schneider (2013) propose a quasi-Gaussian method by applying the Gaussian approximation on an unconstrained variable that is transformed from the constrained correlation functions (Keitel & Schneider 2011;). They find it a better approximation than the ususal Gaussian approximation.…”

CFHTLenS: a Gaussian likelihood is a sufficient approximation for a cosmological analysis of third-order cosmic shear statistics

“…The success will also depend on realistic models of statistical uncertainties in the shear estimators. For this, a Gaussian likelihood is typically used in the statistical analysis, such as in F14, whereas at least for second-order cosmic shear statistics there is evidence in favour of more complex models Keitel & Schneider 2011;Sato et al 2011;Wilking & Schneider 2013). For this paper, we hypothesise that a Gaussian model for the data likelihood of third-order shear correlations possibly yields biased results for cosmological parameters.…”