BAYESIAN MODEL SELECTION Royal Astronomical Society COSMOLOGICAL BAYESIAN MODEL SELECTION Roberto Trotta Oxford Astrophysics & Royal Astronomical Society

Cosmology: a data-driven discipline Cosmic Microwave Background anisotropies 1977 – dipole dT » 3.3 mK COBE 1992 – dT » 18  K on angular scales  > 7° WMAP 2003 – 30 times more resolution,  > 0.2° Large scale structures Gravitational lensing

The need for model selection The spectral index of cosmological perturbation: do we need a spectral tilt ? Harrison-Zel’dovich “scale invariant spectrum” How can we decide whether nS = 1 is “ruled out” ? Can we confirm a prediction of a certain model (eg. that nS = 1) ? Easy ! But... hold on ! Bayesian evidence

Bayes factors for model comparison the evidence of the data in favor of the model the posterior prob’ty of the model given the data (assuming ) 2 competing models Bayes factor : The hitchhiker’s (rough) guide: B01 ln B01 Interpretation < 3 < 1.2 not worth the mention < 10 < 2.3 moderate < 100 < 4.6 strong >100 > 4.6 decisive

Bayes factor as Occam’s razor Model 0 : no free params Model 1: 1 free param Automatic “Occam’s razor” Disfavors complex models, penalizing “wasted” parameter space posterior volume prior structure posterior odds

The number of ’s is not enough A toy example Sampling statistics Your measurement is  sigmas’s away from the value w0 predicted under your model Null hypothesis ( H0 : w = w0 ) testing: Reject the null hypothesis with a certain significance level  E.g. for  = 0.05, we can reject H0 at the 95% confidence level for  > 1.96 w w0

Lindley’s paradox  = 1.96 for all 3 cases but different information content of the data simpler model model with 1 extra parameter

The Savage-Dickey formula How can we compute Bayes factors efficiently ? For nested models and separable priors: use the Savage-Dickey density ratio Model 1 has one extra param than Model 0 no correlations between priors predicted value under Model 0 w0 w prior posterior Economical: at no extra cost than MCMC Exact: no approximations (apart from numerical accuracy) Intuitively easy, clarifies role of prior

Cosmological model selection RT (2005) RT & Melchiorri (2005) The role of the information content Bayes factor B01 information content number of sigma’s Mismatch with prediction ns : scale invariance  : flatness fiso : adiabaticity CNB: neutrino background anisotropies CNB

Expected Posterior Odds RT (2005) ExPO: a new hybrid technique The probability distribution for the model comparison result of a future measurement Conditional on our present knowledge Useful for experiment design & model building: e.g. “Can we confirm that dark energy is a cosmological constant?” Current data posterior ExPO Start from the posterior PDF from current data Fisher Matrix forecast at each sample Combine Laplace approximation & Savage-Dickey formula Compute Bayes factor probability distribution

ExPO: an application Scale invariant vs nS  1 : ExPO for the Planck satellite (2007) About 90% probability that Planck will disfavor nS = 1 with odds of 1:100 or higher

Summary BAYESIAN MODEL SELECTION IN COSMOLOGY Bayesian evidence takes into account the information content of the data Evidence can and does accumulate in favor of models The better your data, the higher the threshold (number of sigma’s away) to reject a prediction Prior choice: a thorny and debatable issue... Savage-Dickey formula for efficient computation of Bayes factors ExPO : Bayes factor forecast conditional on our current knowledge. Research underway to improve on approximations