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

2. 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

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
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

5. 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

6. 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

7. Lindley’s paradox  = 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
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

9. 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

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