Calibrating Bayesian tension statistics using neural ratio estimators

When fits of the same physical model to two different datasets disagree, we call this tension. Several apparent tensions in cosmology have occupied researchers in recent years, and a number of different metrics have been proposed to quantify tension. Many of these metrics suffer from limiting assumptions, and correctly calibrating these is essential if we want to successfully determine whether discrepancies are significant. A commonly used metric of tension is the evidence ratio <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>R</a:mi></a:mrow></a:math>. The statistic has been widely adopted by the community as a Bayesian way of quantifying tensions, however, it has a nontrivial dependence on the prior that is not always accounted for properly. We show that this can be calibrated out effectively with neural ratio estimation. We demonstrate our proposed calibration technique with an analytic example, a toy example inspired by 21-cm cosmology, and with observations of the baryon acoustic oscillations from the Dark Energy Spectroscopic Instrument (DESI) and the Sloan Digital Sky Survey (SDSS). We find no significant tension between DESI and SDSS.