Bayesian evidence & complexity asBayesian evidence & complexity asstatistical tools for dark energystatistical tools for dark energy

Roberto TrottaOxford Astrophysics& Royal Astronomical Society

... work in progress...

The Nature of Dark EnergyThe Nature of Dark Energy

The equation of state parameter w(z) = p/ρ– w = -1– w = const π -1– w(z)– or perhaps another theory of gravity

λ Theoretical explanations must be guided by observationalconstraints:

weff ~ -1 § 0.1

Seljak et al 2005Jarvis et a 2005

DE Reconstruction DE Reconstruction –– PCA PCA

Crittenden & Pogosian 2005 Huterer & Starkman 2003

PCA - biasPCA - bias

Huterer & Starkman 2003

Model Selection ToolboxModel Selection Toolbox

the posterior prob’tyof the model given the data

evidence

ω0 ω

prior

posterior

Automatic Occam’s razor

but useless parameters do not contribute

Model ComparisonModel Comparison

Bayes factor B01

information content

number of sigma’s

Bayes factor

RT (2005), RT & Melchiorri (2005)

CNB

Mism

atch

with

pre

dict

ion

ω0ω

How well must we do?How well must we do?

Q: is DE in form of a cosmological constant?

Model comparison estimation:w = -1 against a fluid-like case -1/3< w < -1

w = -1 measured with an accuracy of gives odds of or evidence5%~1%

0.2%

12:1

60:1300:1

substantial

strong

decisive

Introducing ComplexityIntroducing Complexity

“For every complex problem, there is a solutionthat is simple, neat, and wrong”

Oscar Wilde

How complex a model of w(z) do weneed/can we afford?

Bayesian complexity

Together with Martin Kunz (Geneva) & David Parkinson (Sussex)

Illustration: Linear ModelsIllustration: Linear Models

n-dimensional param spacemeasure everything in units of the prior, P = Id

Evidence: posterior VOLUME

Complexity: Trace of posterior covariance matrix

Application: evidence degeneracy breaking

Looking for structure in the dataLooking for structure in the data

Gull 1988

Example: Polynomial FittingExample: Polynomial Fitting

Data come from a model with N = 6

GOOD DATA, D=10 INSUFFICIENT DATA, D=4

Preliminary DE ResultsPreliminary DE Results

PRESENT SN DATA

n=1: cosmological constantn=2: -1/3 < w = const < -1n=3: 2 w binsn=4: 3 w bins....

complexity

evidence

Summary & Work AheadSummary & Work Ahead

Information-theory approach to the data Complexity: a measure of information content of

the data wrt model Evidence & Complexity: model selection Future work: optimal exploitation of data, binning

issues Beyond SN: BAO, WL,...