#### Transcript Parkinson

```Cosmological Model Selection
David Parkinson
(with Andrew Liddle & Pia Mukherjee)
Outline
• The Evidence: the Bayesian model
selection statistic
• Methods
• Nested Sampling
• Results
Concordance Cosmology
• A flat universe composed of baryons,
cold dark matter and dark energy
• Gaussian, adiabatic and nearly scale
invariant initial perturbations
Model Extensions
• Do we really need only 5 numbers to describe the
universe (b, CDM, H0, As, )?
• Extra dynamic properties: curvature (k), massive
neutrinos (M), dynamic dark energy (w(z)) etc..
• More complex initial conditions: tilt (ns) and
running (nrun) of the adiabatic power spectrum,
entropy perturbations etc..
• How do we decide if these extensions are
justified?
Bayesian Statistics
• Two identical urns A and B
• A contains 99 black balls and 1
white; B has 99 white and 1
black
• P(black|urn A) = 0.99
• Now shuffle the two urns, and
pull out a ball from it. Suppose it
is black. What is the probability
it is from urn A?
• Bayesian statistics allows
probabilities not just of data,
but also parameters and models
Bayes’ Theorem
• Bayes’ theorem gives the posterior probability of
the parameters () of a model (H) given data (D)
P(D | , M)P( | M)
P( | D, M) 
P(D | M)
• Marginalizing over  the evidence is

E=average
P(Dlikelihood
| , M)P(of the
| M)d
 over the
• Evidence
data
 prior parameter space of the model
Jeffrey’s Scale
• The evidence (or model likelihood) updates the prior model
probability through Bayes’ theorem to give the posterior
probability of the model.
• The ratio of two model posteriors is known as the Bayes’ factor:
•
P(D | M1)P(M1) E(M1 )P(M1 )
B10 

Jeffrey’s scale
P(D | M 0 )P(M 0 ) E(M 0 )P(M 0 )

0 < Log B10 < 1
No evidence
1 < Log B10 < 2.5
Weak evidence
2.5 < Log B10 < 5
Strong evidence
Log B10 >5
Decisive evidence
Occam’s Razor
E
 | M)d
 P(D | , M)P(

ˆ, M) 
 P(D | 
Best fit
likelihood

Occam
Factor
• Models are rewarded
for fitting the data well,
and also their
predictive-ness
• Consider three data sets, measuring. By sampling
statistics, all three rule out =0 (the simpler model)
at 95% confidence. But B01=0.5, 1.8, 18 resp. in favor
of the more complex model as the data improves.
Trotta 2007
Methods
• The Laplace Approximation
– Assumes that the P(|D,M) is a multi-dimensional Gaussian
• The Savage-Dickey Density Ratio
– Needs separable priors and nested model
– (and the reference value to be in the high likelihood region of the
more complex model for accuracy)
• Thermodynamic Integration
– Needs a series of MCMC’s at different temperatures
– Accurate but computationally very intensive
• VEGAS
– Likelihood surface needs to be “not too far” from Gaussian
Nested Sampling
Nested Sampling (Skilling 2004/5)
performs the integral 1
L( )P( )d   L(X)dX

using Monte-Carlo samples to trace the
E
0
variation in likelihood with prior mass
(X), and peeling away thin nested isosurfaces of equal likelihood.
• The prior mass is sampled uniformly
• The evidence is incremented using
minimum likelihood point
• Discarding this point reduces X by
a known factor
• A new random point is found with
L > the previous minimum likelihood
Nested Sampling
• Each iteration
reduces X by a
factor N/(N+1) (on
average), where
this factor is the
expectation value of
the largest of N
sampled from
U(0,1).
• The N ‘live’ points
migrate to the high
likelihood regions,
always sampling
uniformly from the
remaining prior
volume (X).
Movie
QuickTime™ and a
H.264 decompressor
are needed to see this picture.
Stopping Criterion
TOTAL
We stop when some accuracy
criterion is met on the sum of the
accumulated evidence from
from the remaining points.
Numerical uncertainty
Calculation proceeds in this direction
is dominated by the Poisson
variability in the number of steps
to reach the posterior H /log( N /N 1)
where H  log( dP /dX)dP
is the logarithm of the compression ratio.

Posterior Samples
The nested sampling algorithm also
generates a set of posterior samples
for parameter estimation:
Li wi
posi 
;
E
wi  12 (X i1  X i1 )
Applications: WMAP3
•
•
•
WMAP alone cannot
distinguish between HZ
and a tilted ( ns) model
Some evidence for ns≠1
from WMAP3+extra, but
only at odds of 8:1.
Datasets
Model
ln B01
WMAP3 only ns (0.8-1.2)
0.34 ± 0.26
WMAP3+ext ns (0.8-1.2)
1.99 ± 0.26
Inflation predicts both
scalar (ns) and tensor (r)
perturbations. HZ is
preferred, unless a log
prior is used on r.
ns+r
(uni: 0-1)
-1.45 ± 0.45
ns+r
(log:-80 - 0)
1.90 ± 0.24
Dark Energy Models
w(a) = w0 + (1-a)wa for z 0 to 2
Models
 ln E
Prob
I:
0.0
63%
II: -1≤ w ≤ -0.33
-1.3 ± 0.1
17%
III: -2 ≤ w ≤ -0.33
-1.8 ± 0.1
10%
IV: -2 ≤ w0 ≤ -0.33,
-2.0 ± 0.1
9%
-4.1 ± 0.1
1%
CDM
-1.33 ≤wa ≤ 1.33
V: -1 ≤w(a) ≤1
Or 78%, 21% and 1% for models I, II & V
Liddle, Mukherjee, Parkinson & Wang 2006
Conclusions
• Model selection (via Bayesian evidences) and
parameter estimation are two levels of inference.
• The nested sampling scheme computes evidences
accurately and efficiently; also gives parameter
posteriors (www.cosmonest.org)
• Applications - simple models still favoured
– model selection based forecasting
– Bayesian model averaging
– many others… Foreground contamination, cosmic
topology, cosmic strings…
```