On the full non-Gaussian Surprise statistic and the cosmological concordance between DESI, SDSS and Pantheon+

With the increasing precision of recent cosmological surveys and the discovery of important tensions within the $\Lambda$CDM paradigm, it is becoming more and more important to develop tools to quantify accurately the discordance between different probes. One such tool is the Surprise statistic, a measure based on the Kullback-Leibler divergence. The Surprise, however, has been up to now applied only under its Gaussian approximation, which can fail to properly capture discordance in cases that deviate significantly from Gaussianity. In this paper we developed the \texttt{klsurprise} code which computes the full numerical non-Gaussian Surprise, and analyse the Surprise for BAO + BBN and supernova data. We test different cosmological models, some of which the parameters deviate significantly from Gaussianity. We find that the non-Gaussianities, mainly present in the Supernova dataset, change the Surprise values significantly from its Gaussian approximation, and reveal a borderline $2.0\sigma$ tension in the curved $w$CDM model (o$w$CDM) between the combined Pantheon+ and SH0ES (Pantheon+&SH0ES) data and the dataset which combines SDSS, BOSS and eBOSS BAO. This modest tension is hidden in the Gaussian Surprise approximation. For DESI, the discrepancy with Pantheon+&SH0ES is at the $1.5\sigma$ level for o$w$CDM, but a large $3.4\sigma$ for $\Lambda$CDM. Removing SH0ES data drops this significance to $2.6\sigma$.