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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°
Gravitational lensing
Large scale structures
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” ?
Easy ! But... hold on !
• Can we confirm a
prediction of a certain
model (eg. that nS = 1) ?
Bayesian evidence
Bayes factors for model comparison
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
posterior volume
posterior odds
prior structure
Automatic
“Occam’s razor”
Disfavors complex
models, penalizing
“wasted” parameter
space
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
w0
w
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
The Savage-Dickey formula
Dickey (1971)
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
posterior
Economical: at no extra cost than
MCMC
Exact: no approximations (apart
from numerical accuracy)
Intuitively easy, clarifies role of prior
prior
w0
w
Cosmological model selection
RT (2005)
RT & Melchiorri (2005)
The role of the information content
Bayes factor B01
Mismatch with prediction
information content
number of sigma’s
ns : scale invariance
CNB
 : flatness
fiso : adiabaticity
CNB: neutrino background anisotropies
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 & SavageDickey 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