Bayesian evidence for the tensor-to-scalar ratio $r$ and neutrino masses
  $m_\nu$: Effects of uniform vs 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 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.
