Applications of Bayesian model selection to cosmological parameters

Abstract

Bayesian model selection is a tool to decide whether the introduction of a new parameter is warranted by data. I argue that the usual sampling statistic significance tests for a null hypothesis can be misleading, since they do not take into account the information gained through the data, when updating the prior distribution to the posterior. On the contrary, Bayesian model selection offers a quantitative implementation of Occam's razor. I introduce the Savage-Dickey density ratio, a computationally quick method to determine the Bayes factor of two nested models and hence perform model selection. As an illustration, I consider three key parameters for our understanding of the cosmological concordance model. By using WMAP 3-year data complemented by other cosmological measurements, I show that a non-scale invariant spectral index of perturbations is favoured for any sensible choice of prior. It is also found that a flat Universe is favoured with odds of 29:1 over non--flat models, and that there is strong evidence against a CDM isocurvature component to the initial conditions which is totally (anti)correlated with the adiabatic mode (odds of about 2000:1), but that this is strongly dependent on the prior adopted. These results are contrasted with the analysis of WMAP 1-year data, which were not informative enough to allow a conclusion as to the status of the spectral index. In a companion paper, a new technique to forecast the Bayes factor of a future observation is presented.

v2 to v3: minor changes, matches accepted version by MNRAS. v1 to v2: major revision. New results using WMAP 3-yr data, scale-invariant spectrum now disfavoured with moderate evidence. New benchmark test for the accuracy of the method. Bayes factor forecast methodology (PPOD, formerly called ExPO) expanded and now presented in a companion paper (astro-ph/0703063)

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