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COSMOLOGICAL BAYESIAN MODEL SELECTION

COSMOLOGICAL BAYESIAN MODEL SELECTION. Roberto Trotta Oxford Astrophysics & Royal Astronomical Society. Cosmology: a data-driven discipline. Large scale structures. Gravitational lensing. Cosmic Microwave Background anisotropies. 1977 – dipole d T » 3.3 mK.

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COSMOLOGICAL BAYESIAN MODEL SELECTION

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

  2. Cosmology: a data-driven discipline Large scale structures Gravitational lensing 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°

  3. 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

  4. Bayes factors for model comparison (assuming ) Model comparison : the evidence of the data in favor of the model the posterior prob’ty of the model given the data 2 competing models Bayes factor : The hitchhiker’s (rough) guide:

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

  6. The number of ’s is not enough 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 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: w w0

  7. Lindley’s paradox Lindley (1954) = 1.96 for all 3 cases but different information content of the data simpler model model with 1 extra parameter

  8. The Savage-Dickey formula 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 posterior Economical: at no extra cost than MCMC Exact: no approximations (apart from numerical accuracy) Intuitively easy, clarifies role of prior prior w0 w Dickey (1971) How can we compute Bayes factors efficiently ?

  9. Cosmological model selection CNB 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

  10. 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

  11. 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

  12. 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 formulafor efficient computation of Bayes factors • ExPO : Bayes factor forecast conditional on our current knowledge. Research underway to improve on approximations

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