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