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Measures of discordance between datasets have become an essential part ofcosmological analyses. It is important to accurately evaluate the significanceof such discordances when present. We propose here a Bayesian interpretation ofinconsistency measures that can extract information about physicalinconsistencies in the presence of data scatter. The new framework is based onthe conditional probability distribution of the level of physical inconsistencygiven the obtained value of the measure. We use the index of inconsistency as acase study to illustrate the new interpretation framework, but this can begeneralized to other metrics. Importantly, there are two aspects in thequantification of inconsistency that behave differently as the number of modelparameters increases. The first is the probability for the level of physicalinconsistency to reach a threshold which drops with the increase of the numberof parameters under consideration. Then the second is the actual level ofphysical inconsistency which remains rather insensitive to such an increase inparameters. This often-overlooked ambiguity leads to an underestimation of thephysical inconsistency by the commonly adopted significance level when used,but is resolved by the Bayesian interpretation because the conditionalprobability distribution includes all the statistical information of the levelof physical inconsistency. Finally, we apply the Bayesian interpretation toexamine the (in)consistency between Planck versus the Cepheid-based localmeasurement, the Dark Energy Survey (DES), the Atacama Cosmology Telescope(ACT) and WMAP. We confirm and revisit the degrees of previous physicalinconsistencies and show the stability of the new interpretation to the numberof cosmological parameters compared to the commonly used n-$\sigma$interpretation when applied to cosmological tensions in multi-parameter spaces.

A Bayesian interpretation of inconsistency measures in cosmology