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 mν , 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 ΛCDM model with odds of about 1 : 20, switching to a logarithmic prior renders both models essentially equally likely. ΛCDM 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.

Improved cosmological constraints on the neutrino mass and lifetime

“…In recent years, a number of studies have attempted to constrain the neutrino mass ordering, showing that under the assumption of stable neutrinos, the inverted ordering is now disfavored by constraints from joint analysis of cosmological and oscillation data [42][43][44][45][46][47] (see also refs. [48][49][50][51] for a different take) as well as from Ly-α observations [52]. However, these arguments are centered on the fact that these analysis lead to a constraint on m ν at odds with the lower bound on the sum of neutrino masses in the case of inverted ordering, m ν 0.1 eV.…”

Cosmological constraints on decaying axion-like particles: a global analysis

“…We can see where the difference in Bayesian evidence arises by using the Occam's razor equation to decompose the evidence into a goodness of fit (average log-likelihood) and Occam penalty (Kullback-Leibler divergence) [125]. By taking logarithms and rearranging Bayes theorem before posterior averaging, we recover the Occams razor equation, which in our case Figure 9 shows that using the full scope of the nested sampling run, we can gain insight into the high likelihood region shown in figure 5.…”

Constraining Quantum Initial Conditions before Inflation

“…For each model the central black line is the calculated log evidence and the coloured bar about it gives the 2σ confidence interval in that value. The dashed lines shows the relative Occam penalty incurred by the vacuum initial condition models, estimated from difference in KL divergence [44] between the model and ΛCDM, calculated using anesthetic [45]. The results show that ΛCDM is significantly better supported by the observations than the vacuum initial condition models, primarily due to the Occam penalty.…”