# Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio

## TL;DR

This paper interprets the Bayesian evidence ratio used to assess tensions between cosmological datasets, proposing a prior-independent modification and introducing a new measure of dataset constraining power.

## Contribution

It introduces a prior-independent Bayesian tension metric using Kullback-Leibler divergences and defines Bayesian model dimensionality for cosmological data analysis.

## Key findings

- DES and Planck are incompatible under LCDM given certain priors.
- SDSS measurements are compatible with Planck, SHOES are not.
- The proposed statistic effectively compares dataset tensions.

## Abstract

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.