Bayesian evidence for the tensor-to-scalar ratio $r$ and neutrino masses $m_ν$: Effects of uniform vs logarithmic priors

Abstract

We review the effect that the choice of a uniform or logarithmic prior has on the Bayesian evidence and hence on Bayesian model comparisons when data provide only a one-sided bound on a parameter. We investigate two particular examples: the tensor-to-scalar ratio $r$ of primordial perturbations and the mass of individual neutrinos $m_ν$, using the cosmic microwave background temperature and polarisation data from Planck 2018 and the NuFIT 5.0 data from neutrino oscillation experiments. We argue that the Kullback-Leibler divergence, also called the relative entropy, mathematically quantifies the Occam penalty. We further show how the Bayesian evidence stays invariant upon changing the lower prior bound of an upper constrained parameter. While a uniform prior on the tensor-to-scalar ratio disfavours the $r$-extension compared to the base LCDM model with odds of about 1:20, switching to a logarithmic prior renders both models essentially equally likely. LCDM with a single massive neutrino is favoured over an extension with variable neutrino masses with odds of 20:1 in case of a uniform prior on the lightest neutrino mass, which decreases to roughly 2:1 for a logarithmic prior. For both prior options we get only a very slight preference for the normal over the inverted neutrino hierarchy with Bayesian odds of about 3:2 at most.