Quantifying tensions in cosmological parameters: Interpreting the DES
  evidence ratio
  We provide a new interpretation for the Bayes factor combination used in the
Dark Energy Survey (DES) first year analysis to quantify the tension between
the DES and Planck datasets. The ratio quantifies a Bayesian confidence in our
ability to combine the datasets. This interpretation is prior-dependent, with
wider prior widths boosting the confidence. We therefore propose that if there
are any reasonable priors which reduce the confidence to below unity, then we
cannot assert that the datasets are compatible. Computing the evidence ratios
for the DES first year analysis and Planck, given that narrower priors drop the
confidence to below unity, we conclude that DES and Planck are, in a Bayesian
sense, incompatible under LCDM. Additionally we compute ratios which confirm
the consensus that measurements of the acoustic scale by the Baryon Oscillation
Spectroscopic Survey (SDSS) are compatible with Planck, whilst direct
measurements of the acceleration rate of the Universe by the SHOES
collaboration are not. We propose a modification to the Bayes ratio which
removes the prior dependency using Kullback-Leibler divergences, and using this
statistical test find Planck in strong tension with SHOES, in moderate tension
with DES, and in no tension with SDSS. We propose this statistic as the optimal
way to compare datasets, ahead of the next DES data releases, as well as future
surveys. Finally, as an element of these calculations, we introduce in a
cosmological setting the Bayesian model dimensionality, which is a
parameterisation-independent measure of the number of parameters that a given
dataset constrains.
