Bayesian evidence & complexity as statistical tools for ...
Transcript of Bayesian evidence & complexity as statistical tools for ...
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,...