Bayesian model choice in cosmology

Bayesian Model Comparison in Cosmology

with Population Monte Carlo
Monthly Notices Royal Astronomical Soc. 405 (4), 2381 - 2390, 2010

Christian P. Robert
Universit´ Paris Dauphine & CREST
e
http://www.ceremade.dauphine.fr/~ xian

Joint works with D., Benabed K., Capp´ O., Cardoso J.F., Fort G., Kilbinger M.,
e
[Marin J.-M., Mira A.,] Prunet S., Wraith D.


Outline

1 Cosmology background

2 Importance sampling

3 Application to cosmological data

4 Evidence approximation

5 Cosmology models

6 lexicon

Cosmology background

Cosmology

A large part of the data to answer some of the major questions in cosmology
comes from studying the Cosmic Microwave Background (CMB) radiation
(fossil heat released circa 380,000 years after the BB).
Huge uniformity of the CMB. Only very sensitive instruments like such as
WMAP (NASA, 2001) can detect ﬂuctuations CMB temperature
e.g minute temperature variations: one part of the sky has a temperature of 2.7251
Kelvin (degrees above absolute zero), while another part of the sky has a temperature
of 2.7249 Kelvin


Cosmology
A large part of the data to answer some of the major questions in cosmology
comes from studying the Cosmic Microwave Background (CMB) radiation
(fossil heat released circa 380,000 years after the BB).
1.0

5
0.8

4
0.6

3
0.4

2
0.2

1
0
0.0

0.0 0.2 0.4 0.6 0.8 1.0 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

CMB

[Marin & CPR, Bayesian Core, 2007]


Plank

Temperature variations are related to ﬂuctuations in the density of
matter in the early universe and thus carry information about the
initial conditions for the formation of cosmic structures such as
galaxies, clusters, and voids for example.
Planck
Joint mission between the European Space Agency (ESA) and NASA, launched in
May 2009. The Planck mission plans to provide datasets of nearly 5 × 1010
observations to settle many open questions with CMB temperature data. Rather than
scalar valued observations, Planck will provide tensor-valued data and thus is likely to
also open up this area of statistical research.


.


Some questions in cosmology

Will the universe expand forever, or will it collapse?
Is the universe dominated by exotic dark matter and what is
its concentration?
What is the shape of the universe?
Is the expansion of the universe accelerating rather than
decelerating?
Is the “ﬂat ΛCDM paradigm” appropriate or is the curvature
diﬀerent from zero?
[Adams, The Guide [a.k.a. H2G2], 1979]


Statistical problems in cosmology
Potentially high dimensional parameter space [Not considered
here]
Immensely slow computation of likelihoods, e.g WMAP, CMB,
because of numerically costly spectral transforms [Data is a
Fortran program]
Nonlinear dependence and degeneracies between parameters
introduced by physical constraints or theoretical assumptions

2.5
0.0

2.0
−1.0
w0

α
1.5
−2.0

1.0
−3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 19.1 19.3 19.5 19.7
Ωm −M

Importance sampling

Importance sampling solutions

1 Cosmology background

2 Importance sampling
Adaptive importance sampling
Adaptive multiple importance sampling

3 Application to cosmological data

4 Evidence approximation

5 Cosmology models

6 lexicon

Importance sampling

Importance sampling 101

Importance sampling is based on the fundamental identity

π(x)
π(f ) = f (x)π(x) dx = f (x) q(x) dx
q(x)

If x1 , . . . , xN are drawn independently from q,
N
π(xn )/q(xn )
π (f ) =
ˆ f (xn )wn ;
¯ wn =
¯ N
,
n=1 m=1 π(xm )/q(xm )

provides a converging approximation to π(f ) (independent of the
normalisation of π).

Importance sampling

Initialising importance sampling
PMC/AIS offers a solution to the difficulty of picking q through
adaptivity:
Given a target π, PMC produces a sequence q t of importance
functions (t = 1, . . . , T ) aimed at approximating π
First sample produced by a regular importance sampling scheme,
x1 , . . . , x1 ∼ q 1 , associated with importance weights
1 N

1 π(x1 )n
wn =
q 1 (x1 )
n

¯1
and their normalised counterparts wn , providing a first
approximation to a sample from π.
Moments of π can then be approximated to construct an updated
importance function q 2 , &c.

Importance sampling


Optimality criterion?

The quality of approximation can be measured in terms of the
Kullback divergence from the target,

π(x)
D(π q t ) = log π(x)dx,
q t (x)

and the density q t can be adjusted incrementally to minimize this
divergence.

Importance sampling

PMC – Some papers

Capp´ et al (2004) - J. Comput. Graph. Stat.
e
Outline of Population Monte Carlo but missed main point
Celeux et al (2005) - Comput. Stat. & Data Analysis Rao-Blackwellisation for
importance sampling and missing data problems
Douc et al (2007) - ESAIM Prob. & Stat. and Annals of Statistics
Convergence issues proving adaptation is positive where q is a mixture density of
random-walk proposals (mixture weights varied)
Capp´ et al (2007) - Stat. & Computing
e
Adaptation of q (mixture density of independent proposals), where weights and
parameters vary
Wraith et al (2009) - Physical Review D
Application of Capp´ et al (2007) to cosmology and comparison with MCMC
e
Beaumont et al (2009) - Biometrika
Application of Capp´ et al (2007) to ABC settings
e
Kilbinger et al (2010) - Month. N. Royal Astro. Soc.
Application of Capp´ et al (2007) to model choice in cosmology
e

Importance sampling

Adaptive importance sampling (2)
Use of mixture densities
D
t t t
q (x) = q(x; α , θ ) = d
t
αt ϕ(x; θd )
d=1
[West, 1993]
where
αt = (αt , . . . , αt ) is a vector of adaptable weights for the D
1 D
mixture components
θ t = (θ1 , . . . , θD ) is a vector of parameters which specify the
t t

components
ϕ is a parameterised density (usually taken to be multivariate
Gaussian or Student-t, the later preferred)

Importance sampling

Capp´ et al (2007) optimal scheme
e

Update q t using an integrated EM approach minimising the KL
divergence at each iteration

π(x)
D(π q t ) = log D
π(x)dx,
t t
d=1 αd ϕ(x; θd )

equivalent to maximising
D
ℓ(α, θ) = log αd ϕ(x; θd ) π(x) dx
d=1

in α, θ.

Importance sampling

PMC updates

Maximization of Lt (α, θ) leads to closed form solutions in
exponential families (and for the t distributions)
For instance for Np (µd , Σd ):

αt+1 =
d ρd (x; αt , µt , Σt )π(x)dx,

xρd (x; αt , µt , Σt )π(x)dx
µt+1 =
d ,
αt+1
d
(x − µt+1 )(x − µt+1 )T ρd (x; αt , µt , Σt )π(x)dx
Σt+1 =
d
d d
.
αt+1
d

Importance sampling

Empirical updates

And empirical versions,
N
αt+1 =
X
d ¯t
wn ρd (xt ; αt , µt , Σt )
n
n=1
PN
wn xt ρd (xt ; αt , µt , Σt )
¯t n n
µt+1 =
d
n=1
αt+1
d
Σt+1
d =
PN
n=1 wn (xt − µt+1 )(xt − µt+1 )T ρd (xt ; αt , µt , Σt )
¯t n d n d n
αt+1
d

Importance sampling

Banana benchmark
2
Twisted Np (0, Σ) target with Σ = diag(σ1 , 1, . . . , 1), changing the
2 2
second co-ordinate x2 to x2 + b(x1 − σ1 )

20
10
0
−10
x2

−20
−30
−40

−40 −20 0 20 40

x1

2
p = 10, σ1 = 100, b = 0.03
[Haario et al. 1999]

Importance sampling

Simulation

20

20
10

10
0

0
−20

−20
−40

−40
−40 −20 0 20 40 −40 −20 0 20 40
20

20
10

10
0

0
−20

−20
−40

−40
−40 −20 0 20 40 −40 −20 0 20 40
20

20
10

10
0

0
−20

−20
−40

−40

−40 −20 0 20 40 −40 −20 0 20 40

Importance sampling

Monitoring by perplexity

Stop iterations when further adaptations do not improve D(π q t ).
The transform exp[−D(π q t )] may be estimated by the normalised
perplexity p = exp(Ht )/N, where
N

N
Ht = −
N ¯t ¯t
wn log wn
n=1

is the Shannon entropy of the normalised weights
Thus, minimization of the Kullback divergence can be
approximately connected with the maximization of the perplexity
(normalised) (values closer to 1 indicating good agreement
between q and π).

Importance sampling

Monitoring by ESS

A second criterion is the eﬀective sample size (ESS)

N −1
2
ESSt
N = ¯t
wn
n=1

which can be interpreted as the number of equivalent iid sample
points.

Importance sampling

Simulation

0.8
0.6
NPERPL

0.4
0.2
0.0

1 2 3 4 5 6 7 8 9 10
0.8
0.6
NESS

0.4
0.2
0.0

1 2 3 4 5 6 7 8 9 10

Normalised perplexity (top panel) and normalised eﬀective sample size(ESS/N) (bottom panel) estimates for the
ﬁrst 10 iterations of PMC

Importance sampling

Comparison to MCMC
Adaptive MCMC: Proposal is a multivariate Gaussian with Σ
updated/based on previous values in the chain. Scale and update
times chosen for optimal results.
PMC MCMC
fa fa

fb fb

Evolution of π(fa ) (top panels) and π(fb ) (bottom panels) from 10k points to 100k points for both PMC (left
panels) and MCMC (right panels).

Importance sampling

Simulation

0.74
fc fe fh

Propoportion of points inside

0.70
0.66
0.62

PMC
d10 PMC MCMC
d10 MCMC PMC
d2 PMC MCMC
d2 MCMC PMC
d1 PMC MCMC
d1 MCMC
1.00

fd fg fi
Propoportion of points inside

0.96
0.92
0.88

PMC
d10 PMC MCMC
d10 MCMC PMC
d2 PMC MCMC
d2 MCMC PMC
d1 PMC MCMC
d1 MCMC

Results showing the distributions of the PMC and the MCMC estimates. All estimates are based on 500 simulation
runs.

Importance sampling


Full recycling:
At iteration t, design a new proposal qt based on all previous
samples
x1 , . . . , x1 , . . . , xt−1 , . . . , xt−1
1 N 1 N

At each stage, the whole past can be used: if un-normalised
weights ωi,t are preserved along iterations, then all xt ’s can be
i
pooled together

Importance sampling

Caveat

When using several importance functions at once, q0 , . . . , qT , with
samples x0 , . . . , x0 0 , . . ., xT , . . . , xT T and importance weights
1 N 1 N
t
ωi = π(xt )/qt (xt ), merging thru the empirical distribution
i i

t t
ωi δxt (x)
i
ωi ≈ π(x)
t,i t,i

Fails to cull poor proposals: very large weights do remain large in
the cumulated sample and poorly performing samples
overwhelmingly dominate other samples in the ﬁnal outcome.
c Raw mixing of importance samples may be harmful, compared
with a single sample, even when most proposals are eﬃcient.

Importance sampling

Deterministic mixtures

Owen and Zhou (2000) propose a stabilising recycling of the
weights via deterministic mixtures by modifying the importance
density qt (xt ) under which xt was truly simulated to a mixture of
i i
all the densities that have been used so far
T
1
T
Nt qt (xT ) ,
i
j=0 Nj t=0

resulting into the deterministic mixture weight
T
t 1
ωi = π(xt )
i T
Nt qt (xt ) .
i
j=0 Nj t=0

Importance sampling

Unbiasedness

Potential to exploit the most eﬃcient proposals in the sequence
Q0 , . . . , QT without rejecting any simulated value nor sample.
Poorly performing importance functions are simply eliminated
through the erosion of their weights
T
1
π(xt )
i T
Nl ql (xt )
i
j=0 Nj l=0

as T increases.
Paradoxical feature of competing acceptable importance weights
for the same simulated value well-understood in the cases of
Rao-Blackwellisation and of Population Monte Carlo. More
intricated here in that only unbiasedness remains [fake mixture]

Importance sampling

AMIS

AMIS (or Adaptive Multiple Importance Sampling) uses
importance sampling functions (qt ) that are constructed
sequentially and adaptively, using past t − 1 weighted samples.
i weights of all present and past variables xl
i
(1 ≤ l ≤ t , 1 ≤ j ≤ Nt ) are modiﬁed, based on the current
proposals
ii the entire collection of importance samples is used to build
the next importance function.
[Parallel with IMIS: Raftery & Bo, 2010]

Importance sampling

The AMIS algorithm

Adaptive Multiple Importance Sampling
At iteration t = 1, . . . , T
ˆ
1) Independently generate Nt particles xt ∼ q(x|θ t−1 )
i
t ˆl−1
2) For 1 ≤ i ≤ Nt , compute the mixture at xi δi = N0 q0 (xt ) +
t Pt
ﬃ t i l=1 Nl q(xi ; θ ) and derive the
t t t ‹ t Pt
weight of xi , ωi = π(xi ) [δi {N0 + l=0 Nl }] .

3) For 0 ≤ l ≤ t − 1 and 1 ≤ i ≤ Nl , actualise past weights as

t
l l l ˆt−1 l l ‹ l‹
X
δi = δi + q(xi ; θ ) and ωi = π(xi ) [δi {N0 + Nl }] .
l=0

ˆ
4) Compute the parameter estimate θ t based on

0 0 0 0 t t t t
({x1 , ω1 }, . . . , {xN , ωN }, . . . , {x1 , ω1 }, . . . , {xN , ωN })
0 0 t t

[Cornuet, Marin, Mira & CPR, 2009, arXiv:0907.1254]

Importance sampling

Studentised AMIS

When the proposal distribution qt is a Student’s t proposal,

T3 (µ, Σ)

mean µ and covariance Σ parameters can be updated by
estimating ﬁrst two moments of the target distribution Π
Pt PNl l l Pt PNl
l=0 i=1 ω x ˆ l=0 ωi (xl − µt )(xl − µt )T
l ˆ ˆ
µt = Pt PN i i
ˆ and Σt = i=1 i
Pt PNl l
i
.
l l
l=0 i=1 ωi l=0 i=1 ωi

i.e. using optimal update of Capp´ et al. (2007)
e

Obvious extension to mixtures [and again optimal update of Capp´
e
et al. (2007)]

Importance sampling

Simulations
Same banana benchmark
Target function p AMIS Capp´’07
e
5 0.06558 0.06879
E(x1 ) = 0 10 0.06388 0.11051
20 0.09167 0.17912
5 0.10215 0.11583
E(x2 ) = 0 10 0.21421 0.22557
20 0.25316 0.29087
P5
E(xi ) = 0 5 0.00478 0.00927
Pi=3
10
E(xi ) = 0 10 0.00902 0.02099
Pi=3
20
i=3 E(xi ) = 0 20 0.01666 0.04208
5 2.60672 3.92650
var(x1 ) = 100 10 7.06686 7.48877
20 8.20020 9.71725
5 2.10682 2.96132
var(x2 ) = 19 10 3.76660 5.08474
20 4.85407 5.98031
P5
var(xi ) = 3 5 0.00645 0.01196
Pi=3
10
i=3 var(xi ) = 8
P20
10 0.01370 0.02636
i=3 var(xi ) = 18 20 0.04609 0.06424

Root mean square errors calculated over 10 replications for diﬀerent target functions
and dimensions p.

Importance sampling

Simulation (cont’d)

10 replicate ESSs for AMIS (left) and PMC (right) for p = 5, 10, 20.

Importance sampling


10 replicate absolute errors associated to the estimations of E(x1 ) (left column),
E(x2 ) (center column) and p E(xi ) (right column) using AMIS (left in each
P
i=3
block) and PMC (right) for p = 5, 10, 20.

Importance sampling


10 replicate absolute errors associated to the estimations of var(x1 ) (left column),
var(x2 ) (center column) and p var(xi ) (right column) using AMIS (left in each
P
i=3
block) and PMC (right) for p = 5, 10, 20.

Application to cosmological data

Cosmological data

Posterior distribution of cosmological parameters for recent
observational data of CMB anisotropies (diﬀerences in temperature
from directions) [WMAP], SNIa, and cosmic shear.
Combination of three likelihoods, some of which are available as
public (Fortran) code, and of a uniform prior on a hypercube.


Cosmology parameters
Parameters for the cosmology likelihood
(C=CMB, S=SNIa, L=lensing)

Symbol Description Minimum Maximum Experiment
Ωb Baryon density 0.01 0.1 C L
Ωm Total matter density 0.01 1.2 C S L
w Dark-energy eq. of state -3.0 0.5 C S L
ns Primordial spectral index 0.7 1.4 C L
∆2R Normalization (large scales) C
σ8 Normalization (small scales) C L
h Hubble constant C L
τ Optical depth C
M Absolute SNIa magnitude S
α Colour response S
β Stretch response S
a L
b galaxy z-distribution ﬁt L
c L

For WMAP5, σ8 is a deduced quantity that depends on the other parameters


Adaptation of importance function


Estimates
Parameter PMC MCMC

Ωb 0.0432+0.0027
−0.0024 0.0432+0.0026
−0.0023
Ωm 0.254+0.018
−0.017
0.253+0.018
−0.016
τ 0.088+0.018
−0.016 0.088+0.019
−0.015
+0.059
w −1.011 ± 0.060 −1.010−0.060
ns 0.963+0.015
−0.014 0.963+0.015
−0.014
109 ∆2
R 2.413+0.098
−0.093 2.414+0.098
−0.092
h 0.720+0.022
−0.021 0.720+0.023
−0.021
a 0.648+0.040
−0.041 0.649+0.043
−0.042
b 9.3+1.4
−0.9 9.3+1.7
−0.9
c 0.639+0.084
−0.070 0.639+0.082
−0.070
+0.029
−M 19.331 ± 0.030 19.332−0.031
α 1.61+0.15
−0.14 1.62+0.16
−0.14
−β −1.82+0.17
−0.16
−1.82 ± 0.16
σ8 0.795+0.028
−0.030 0.795+0.030
−0.027

Means and 68% credible intervals using lensing, SNIa and CMB


Advantage of AIS and PMC?

Parallelisation of the posterior calculations
- For the cosmological examples, we used up to 100 CPUs on a computer cluster to explore the cosmology
posteriors using AIS/PMC. Reducing the computational time from several days for MCMC to a few hours
using PMC.

Low variance of Monte Carlo estimates
- For PMC and q closely matched to π, signiﬁcant reductions in the variance of the Monte Carlo
estimates are possible compared to estimates using MCMC. Also translating into a computational saving,
with further savings possible by combining samples across iterations

Simple diagnostics of ‘convergence’ (perplexity)
- For PMC, the perplexity provides a relatively simple measure of sampling adequacy to the target density
of interest

Evidence approximation

Evidence/Marginal likelihood/Integrated Likelihood ...
Central quantity of interest in (Bayesian) model choice

π(x)
E= π(x)dx = q(x)dx.
q(x)

expressed as an expectation under any density q with large enough
support.
Importance sampling provides a sample x1 , . . . xN ∼ q and
approximation of the above integral,
N
E≈ wn
n=1

π(xn )
where the wn = q(xn ) are the (unnormalised) importance weights.


Back to the banana ...

Centred d-multivariate normal, x ∼ Nd (0, Σ) with covariance
2
Σ = diag(σ1 , 1, . . . , 1), which is slightly twisted in the ﬁrst two
1
2
dimensions by changing x2 to be x2 + β(x2 − σ1 ). where σ1 = 100 2

and β controls the degree of curvature.
We integrate over the unormalised target density

E= π(β)f (x|β, Σ) dβ

or
E= π(x|β, Σ) dx.


Simulation results (1)

10

10
0

0
−10

−10
x2

x2
−20

−20
−30

−30
−40 −20 0 20 40 −40 −20 0 20 40

x1 x1
0.03004

−264.028
Posterior mean of β

Evidence (log)
0.03000

−264.032
0.02996

−264.036
0.02992

After 10th iteration After 10th iteration

β unknown


Simulation results (2)

0.8
0.8

0.6
0.6
Perplexity

NESS
0.4
0.4

0.2
0.2

0.0
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Iteration Iteration

Evidence (log): final sample
0.2

0.015
Evidence (log)
0.1

0.005
0.0

−0.015 −0.005
−0.1

1 2 3 4 5 6 7 8 9 10

Iteration After 10th iteration

β = 0.015 known

Cosmology models

Back to cosmology questions

Standard cosmology successful in explaining recent observations,
such as CMB, SNIa, galaxy clustering, cosmic shear, galaxy cluster
counts, and Lyα forest clustering.
Flat ΛCDM model with only six free parameters
(Ωm , Ωb , h, ns , τ, σ8 )
Extensions to ΛCDM may be based on independent evidence
(massive neutrinos from oscillation experiments), predicted by
compelling hypotheses (primordial gravitational waves from
inflation) or reflect ignorance about fundamental physics
(dynamical dark energy).
Testing for dark energy, curvature, and inflationary models

Cosmology models

Extended models

Focus on the dark energy equation-of-state parameter, modeled as

w = −1 ΛCDM
w = w0 wCDM
w = w0 + w1 (1 − a) w(z)CDM

In addition, curvature parameter ΩK for each of the above is either
ΩK = 0 (‘ﬂat’) or ΩK = 0 (‘curved’).
Choice of models represents simplest models beyond a
“cosmological constant” model able to explain the observed,
recent accelerated expansion of the Universe.

Cosmology models

Cosmology priors

Prior ranges for dark energy and curvature models. In case of w(a)
models, the prior on w1 depends on w0
Parameter Description Min. Max.
Ωm Total matter density 0.15 0.45
Ωb Baryon density 0.01 0.08
h Hubble parameter 0.5 0.9
ΩK Curvature −1 1
w0 Constant dark-energy par. −1 −1/3
w1 Linear dark-energy par. −1 − w0 −1/3−w0
1−aacc

Cosmology models

Cosmology priors (2)

Component to the matter-density tensor with w(a) < −1/3 for
values of the scale factor a > aacc = 2/3. To limit the state
equation from below, we impose the condition w(a) > −1 for all a,
thereby excluding phantom energy.
Natural limit on the curvature is that of an empty Universe, i.e.
upper boundary on the curvature ΩK = 1. A lower boundary
corresponds to an upper limit on the total matter-energy density:
ΩK > −1, excluding high-density Universe(s) which are ruled out
by the age of the oldest observed objects.
Alternative prior on ΩK could be derived from the paradigm of inﬂation, but most
scenarios imply the curvature to be , on the order of 10−60 . The likelihood over such
a prior on ΩK is essentially ﬂat for any current and future experiments, hence cannot
be assessed.

Cosmology models

PMC setup
q 0 is a Gaussian mixture model with D components randomly
shifted away from the MLE and covariance equal to the
information matrix.
For the dark-energy and curvature models number of
iterations T equal to 10, unless perplexity indicated the
contrary. Average number of points sampled under an
individual mixture-component, N/D, controlled for stable
updating component (N = 7 500 and D = 10).
For the primordial models T = 5, N = 10 000 and D between
7 and 10, depending on the dimensionality.
Parameters controlling the initial mixture means and
covariances, chosen as fshift = 0.02, and fvar between 1 and
1.5. Final iteration run with a ﬁve-times larger sample

Cosmology models

Results

In most cases evidence in favour of the standard model. especially
when more datasets/experiments are combined.
Largest evidence is ln B12 = 1.8, for the w(z)CDM model and
CMB alone. Case where a large part of the prior range is still
allowed by the data, and a region of comparable size is excluded.
Hence weak evidence that both w0 and w1 are required, but
excluded when adding SNIa and BAO datasets.
Results on the curvature are compatible with current ﬁndings:
non-ﬂat Universe(s) strongly disfavoured for the three dark-energy
cases.

Cosmology models

Evidence
Evidence (reference model ΛCDM flat)

CMB

mod.
4 CMB+SN
CMB+SN+BAO

weak inconcl. weak
2 w(z) flat
w(z) flat
w0 flat
0
w(z) flat
w(z) curved
ln B12

-2 Λ curved
w0 flat
w0 curved
mod.

-4 w(z) curved
Λ curved w(z) curved
Λ curved
strong

-6

w0 curved
-8 w0 curved

4 5 6
npar

Cosmology models

Posterior outcome
Posterior on dark-energy parameters w0 and w1 as 68%- and 95% credible regions for
WMAP (solid blue lines), WMAP+SNIa (dashed green) and WMAP+SNIa+BAO
(dotted red curves). Allowed prior range as red straight lines.

2.0

1.5

1.0
w1

0.5

0.0

−0.5
−1.0 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4
w0

Cosmology models

PMC stability

wCDM flat wCDM curvature
−8.5

−10
−9.0

−11
−9.5
ln E

ln E
−10.0

−12
−13
−11.0

−14
1 2 3 4 5 6 7 8 9 10 1 3 5 7 9 11 13 15 17 19

iteration iteration

Distribution of 25 PMC samplings of two dark-energy models, ﬂat wCDM (left panel)
and curved wCDM (right panel). Log-evidence

Cosmology models

PMC stability

wCDM flat wCDM curvature

0.5
0.8

0.4
0.6
perplexity

perplexity

0.3
0.4

0.2
0.2

0.1
0.0

0.0
1 2 3 4 5 6 7 8 9 10 1 3 5 7 9 11 13 15 17 19

iteration iteration

Distribution of 25 PMC samplings of two dark-energy models, ﬂat wCDM (left panel)
and curved wCDM (right panel). Perplexity

lexicon

lexicon

BAO, baryon acoustic oscillations
CMB, cosmic microwave background radiation
COBE, cosmic background explorer
ΛCDM, lambda-cold dark matter
Lyα, Lyman-alpha
SNIa, type Ia supernovae
WMAP, Wilkinson microwave anisotropy probe

Bayesian model choice in cosmology

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