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We investigate whether a Gaussian likelihood, as routinely assumed in theanalysis of cosmological data, is supported by simulated survey data. We definetest statistics, based on a novel method that first destroys Gaussiancorrelations in a dataset, and then measures the non-Gaussian correlations thatremain. This procedure flags pairs of datapoints which depend on each other ina non-Gaussian fashion, and thereby identifies where the assumption of aGaussian likelihood breaks down. Using this diagnostic, we find thatnon-Gaussian correlations in the CFHTLenS cosmic shear correlation functionsare significant. With a simple exclusion of the most contaminated datapoints,the posterior for $s_8$ is shifted without broadening, but we find nosignificant reduction in the tension with $s_8$ derived from Planck CosmicMicrowave Background data. However, we also show that the one-pointdistributions of the correlation statistics are noticeably skewed, such thatsound weak lensing data sets are intrinsically likely to lead to asystematically low lensing amplitude being inferred. The detectednon-Gaussianities get larger with increasing angular scale such that for futurewide-angle surveys such as Euclid or LSST, with their very small statisticalerrors, the large-scale modes are expected to be increasingly affected. Theshifts in posteriors may then not be negligible and we recommend that thesediagnostic tests be run as part of future analyses.

On the insufficiency of arbitrarily precise covariance matrices: non-Gaussian weak lensing likelihoods