Applications of Bayesian Model Selection to Cosmological Parameters
Abstract
Bayesian evidence is a tool for model comparison which can be used to decide whether the introduction of a new parameter is warranted by data. I show that the usual sampling statistic rejection tests for a null hypothesis can be misleading, since they do not take into account the information content of the data. I review the Laplace approximation and the Savage-Dickey density ratio to compute Bayes factors, which avoid the need of carrying out a computationally demanding multi-dimensional integration. I present a new procedure to forecast the Bayes factor of a future observation by computing the Expected Posterior Odds (ExPO). As an illustration, I consider three key parameters for our understanding of the cosmological concordance model: the spectral tilt of scalar perturbations, the spatial curvature of the Universe and a CDM isocurvature component to the initial conditions which is totally (anti)correlated with the adiabatic mode. I find that current data are not informative enough to draw a conclusion on the status of the spectral tilt, while there is moderate evidence for a flat Universe (with odds of order 10:1) and strong evidence for purely adiabatic initial conditions (odds larger than 1000:1). I show that the Planck satellite has about 90% of probability of gathering large evidence against a scale invariant spectral index, while a clear cut model selection for a flat universe will still require the use of data complementary to CMB measurements. The issue of priors in Bayesian model selection is also discussed, and it is shown that in a phenomenological modelling the prior for a given experiment can be computed by sensitivity analysis.Keywords
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