Bayesian evidence for the tensor-to-scalar ratio r and neutrino masses mν: Effects of uniform versus logarithmic priors

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(nu), using the cosmic microwave background temperature and polarization 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 disfavors the r extension compared to the base Lambda CDM model with odds of about 1:20, switching to a logarithmic prior renders both models essentially equally likely. Lambda CDM with a single massive neutrino is favored 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.

Automated summary

The paper examines the impact of choosing uniform or logarithmic priors on Bayesian evidence and model comparisons in situations where data only provide a one-sided bound on a parameter. Using specific examples regarding the tensor-to-scalar ratio and individual neutrino mass, the study demonstrates the mathematical quantification of Occam's penalty through the Kullback-Leibler divergence. The results show invariance of Bayesian evidence when changing the prior bounds of constrained parameters, and highlight the different preferences of models depending on the choice of prior.