Transcript Global Fits: Making sense out of what we will see

Global Fits:
Making Sense out of What we Will See
Oliver Buchmüller
Roberto Trotta
Imperial College London
•Global Fits – Motivation & History
• “SUSY Fits” – where are we today?
•The future
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
“Random Scans”: Why we Need Better Tools
Gogoladze et al (2008) [one of
many such studies]
• Points accepted/rejected in a in/out fashion (e.g., 2-sigma cuts)
• No statistical measure attached to density of points: no probabilistic interpretation
of results possible
• Inefficient/Unfeasible in high dimensional parameters spaces (N>3)
• Explores only a very limited portion of the parameter space!
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Statistics in High-D spaces
• “Random scans” of a high-dimensional parameter space only probe a very
limited sub-volume: this is the concentration of measure.
• Statistical fact: the norm of d draws from U[0,1] concentrates around (d/3)1/2
with constant variance
Tom Loredo (cosmostats09 talk)
Damien Francois (2005)
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Geometry in High-D Spaces
• Geometrical fact: in d dimensions, most of the volume is near the boundary.
The volume inside the spherical core of d-dimensional cube is negligible.
Volume of cube
1
Volume of sphere
1
Ratio Sphere/Cube
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Scanning the New Physics Parameter Space
Determining the preferred NP parameter space is a multi-dimensional problem. Even in one of
the simplest cases, the CMSSM, there are four NP parameters (M0, M1/2, A0, tanβ) as well as
SM parameters like mtop or mb. The conventional (now historical) strategy in the field was to carry
out “2D scans” by fixing the other relevant parameters to certain values.
An arbitrary example: 0106334 [hep-ph]
tanβ=10
tanβ=50
2D scans strongly depend on the “fixed NP parameters”
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Scanning the New Physics Parameter Space
Mtop=170 GeV
Mtop=180 GeV
There is also a strong dependence on the important SM
parameters! (which are known only with a limited accuracy)
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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The Solution to the Problem – Global Fits
Carry out a simultaneous fit
of all relevant NP and SM
parameter to the
experimental
data/constraints.
Valid for all NP and SM parameters
Marginalize (= integrate) or
maximise along the hidden
dimensions so as to obtain
results that account for the
multi-dimensional nature of
the problem.
This gives a statistically welldefined answer.
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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The Solution to the Problem – Global Fits
Another Example: the so-called “WMAP strips”
WMAP
strips
a few
years ago
Wmap
blobs
today
In 2D scans, enforcing the
cosmological relic
abundance results in narrow
“allowed regions” (the
“WMAP strips”), whose
location changes with the
value of the fixed
parameters.
Once fixed parameters are
included and hidden
dimensions accounted for,
WMAP strips widen to
become “WMAP blobs”
0306219 [hep-ph]
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Different Statistical Approaches
Bayesian
vs
Frequentist
Different answers to different questions
ρbar = 0.155 [+0.022 -0.022]
ηbar = 0.342[+0.014 -0.014]
ρbar = 0.139 [+0.025 -0.027]
ηbar = 0.341[+0.016 -0.015]
CKM triangle Fits Today:
Overall good agreement between the two approaches but still differences in the details.
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Marginal Posterior vs Profile Likelihood
Bayesian
vs
Marginal posterior:
Frequentist
Profile likelihood:
Physical analogy:
(thanks to Tom Loredo)
Heat:
Posterior:
Relevant for
underconstrained system
θ2
Posterior =
region with most heat
likelihood =
“hottest” hypothesis
(plot depicts likelihood contours - prior assumed flat over wide range)
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
θ1
Bayesian Statistic on the Rise?
As far as I can think back, the frequentist approach has dominated the field of
particle physics – it still does but bayesian methodology becomes increasingly
more important it seems. In particular there is strong influence from the field
of astrophysics:
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Global NP Fits in the LHC (Discovery) Era
R. Lafaye , M. Rauch, T. Plehn, D. Zerwas
H. Flächer, M. Goebel, J. Haller,
A. Höcker, K. Mönig, J. Stelzer
P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann
GFitter
Sfitter
Fittino
L. Roszkowski, R. Ruiz de Austri, R. Trotta
O. Buchmueller, R. Cavanaugh, A. De Roeck, J.R. Ellis,
H.Flacher, S. Heinemeyer, G. Isidori, K.A. Olive,
F.J. Ronga, G. Weiglein
S.S. AbdusSalam, B.C. Allanach, M.J. Dolan,
F. Feroz, M.P. Hobson
MasterCode
Superbayes
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Global NP Fits in the LHC (Discovery) Era
R. Lafaye , M. Rauch, T. Plehn, D. Zerwas
H. Flächer, M. Goebel, J. Haller,
A. Höcker, K. Mönig, J. Stelzer
P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann
GFitter
Sfitter
Fittino
Many different groups of theorists and experimentalists
and several different approaches!
L. Roszkowski, R. Ruiz de Austri, R. Trotta
O. Buchmueller, R. Cavanaugh, A. De Roeck, J.R. Ellis,
H.Flacher, S. Heinemeyer, G. Isidori, K.A. Olive,
F.J. Ronga, G. Weiglein
S.S. AbdusSalam, B.C. Allanach, M.J. Dolan,
F. Feroz, M.P. Hobson
MasterCode
Superbayes
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Global NP Fit Methodology in a Nutshell
• Chose a NP model
• SUSY as show-case – so far considered models:
•
•
GUT Scale model: CMSSM, NUHM, AMSB
Soft Scale model: pMSSM
• Chose the measurements
• Considered (indirect) constraints are collider (EWK, flavour) and noncollider data (g-2, relic density)
•
It is crucial to have a consistent set of calculations for these constraints!
• Chose your preferred statistical approach to confront
model prediction Pi with measurements Mi with cov(Mi,Mi)
• “χ2 “ as an example: (M – P)T cov-1 (M – P) [+ limits]
•
Find set of Pi that minimize χ2
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Constrained MSSM Analysis Pipeline
SCANNING ALGORITHM
Likelihood = 0
4 CMSSM parameters
θ = {m0, m1/2, A0, tanβ}
(fixing sign(μ) > 0)
NO
RGE
4 SM “nuisance
parameters”
Ψ={mt, mb,αS, αEM }
Data:
Gaussian likelihoods
for each of the Ψj
(j=1...4)
Non-linear
numerical
function
via SoftSusy 2.0.18
DarkSusy 4.1
MICROMEGAS 2.2
FeynHiggs 2.5.1
Hdecay 3.102
Observable
quantities
fi(θ ,Ψ)
CDM relic abundance
BR’s
EW observables
g-2
Higgs mass
sparticle spectrum
(gamma-ray, neutrino,
antimatter flux, direct
detection x-section)
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Physically acceptable?
EWSB, no tachyons,
neutralino CDM
YES
Joint likelihood function
Data:
Gaussian likelihood
(CDM, EWO, g-2, b→sγ, ΔMBs)
other observables have
only lower/upper limits
Finding the Favoured Regions
• Due to the weak nature of constraints, different scanning
techniques and statistical methods will generally give
different answers (also because the questions being asked
are different!)
• Traditional method: determine best fit parameter (find
minimum)
•
•
•
•
Markove Chain Monte Carlos (MCMC)
MCMC and Minuit as “afterburner”
Simulated annealing
Genetic algorithm
• Determine errors:
Local Δ(LogLikelihood)
• Intelligent sampling of parameter space with MCMC
• Pseudo Experiment/Toy Experiment sampling
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Finding the Favoured Regions
• Alternatively, one might focus on the probability mass
instead (Bayesian)
• Best fit has no special status: look for the bulk of the
posterior probability mass instead
• Markov Chain Monte Carlo techniques (MCMC)
• Nested sampling
• Hamiltonian MC
• Determine errors: region of parameter space containing
e.g. 95% of samples
• Might depend on measure chosen (prior)
• Represents degree of knowledge
(rather than coverage)
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Today: “Weak Constraints” only
Biggest problem today – no significant deviation from the SM!
Only ~2 to ~3σ evidence and relic density that cannot be explained in the
SM.
only weakly constraint NP parameter space
The main players today
Δaμ =(24.6 ± 8)E-10
Mw =80.399 ± 0.023 GeV
Mw
Ωh2 =0.1099 ± 0.0066 ± 0.012
Relic
density
g-2
R(BXsγ) = 1.117±0.12
R(Bτν) = 1.43±0.43
R: Measurement/SM
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Where are we today?
Details in Frederic Ronga’s and Roberto Ruiz
de Austri’s talks
Here just a few points...
CMSSM Today: Frequentist vs. Bayesian
0809.3792 [hep-ph]
0705.0487 [hep-ph]
“naturaless
prior”
“flat prior”
0808.4128 [hep-ph]
“log prior”
All priors find a “low mass SUSY”
solution similar to the prior independent
frequentist result.
The key question is how constrained is
actually this region?
Has the frequentist approach missed
“high mass” regions?
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Prior Dependence
0809.3792 [hep-ph]
“flat prior”
0705.0487 [hep-ph]
“flat prior”
“naturalness
“flat prior”
“naturalness
prior”
“flat prior”
prior”
“log prior”
“log prior”
Prior dependence of the Bayesian fits results from weak
constraints on parameter space. Stronger assumptions (e.g. naturalness priors)
lead to posteriors dominated by prior information (rather than data).
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Prior Dependence due to Weak Constraints
0705.0487 [hep-ph]
Global Fit to CMSSM
with four parameters:
M0, M1/2, A0, tanβ [μ>0]
strong prior
dependence
“naturalness
“flat prior”
prior”
Four (CMSSM) vs. two (LVS) free parameters.
“Volume” effects coming from large-D parameter spaces are
presently important. The question will remain in the future: i.e.,
once we’ll have LHC data, the frontier
will
shift to models in higher
0806.1184
[hep-ph]
Global Fit to Large
dimensions
(eg, pMSSM with ~ 20 free parameters)
Volume String
scenario
with only two param.:
M0, tanβ [μ>0]
smaller prior
dependence
“flat prior”
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
“naturalness
prior”
29
LHC Data (Discoveries) will help a lot 
If we are really luck we might see these
spectacular signatures already very early
at the LHC!
Assumed ATLAS
covariance
matrix for the
SU3 benchmark
Point at 1/fb
Assuming such a spectacular discovery as
input reduces the prior dependence of the
global bayesian fits to almost negligible levels.
“flat prior”
Joint HEP-APP IOP meeting on SUSY - March0907.0594
24th 2010 [hep-ph]
M(l+l-) GeV
“log prior”
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Accelerated Inference from Neural Networks
We will need faster methods to deal with more complex models and higher
dimensional parameter spaces, independently of the statistics used.
Neural networks provide a way forward for ultra-fast inference.
Simulated ATLAS data with
1/fb luminosity
Computational effort for an 8parameters global CMSSM fit:
full
MultiNest
fit
• Standard MCMC
(SuperBayeS v1.23, 2006)
500 CPU days
• MultiNest algorithm
(SuperBayeS v1.35, 2008)
<3 CPU days
speed-up factor: 200
Neural
Network
• SuperBayeS+Neural Networks
(Bridges, RT et al, upcoming)
15 CPU minutes
speed-up factor: 50’000
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Connection to Astro(particle)physics
Direct Dark Matter Searches & LHC
0809.3792 [hep-ph]
Sensitivity Plot:
WIMP(LSP) Mass vs. pSI
pSI: spin-independent dark matter WIMP
elastic scattering cross section on a free
proton.
A convenient way to illustrate direct
and indirect WIMP searches.
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
0907.4468 [hep-ph]
54
Direct Detection Prospects in the CMSSM
Prospects for WIMP discovery in the CMSSM framework for upcming 1t
scale detectors are robust, independently of the choice of statistics.
Bayesian posterior
Profile likelihood
68%
68%
95%
95%
Predicted reach with 1 tonne detectors
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
The Need for a Multiple Probes Approach
• No single probe can cover the whole
favoured parameter space, not even
the LHC.
• Astroparticle probes (direct and
indirect detection) can increase the
coverage of the favoured parameter
space, and deliver increased
statistical robustness.
• High complementarity with direct
detection methods.
Need to establish a common
language (and approach) to
Use all this information consistently
Global Fits are one possible tool for this.
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Cumberbatch et al (in prep)
Conclusions
•Global fits are needed as tool to interpret new physics discoveries from
multiple probes.
•Can be used to explore connections between particle and astroparticle
physics as well as cosmology.
•Two independent statistical approaches: ought to give the same result
(eventually, when data are constraining enough!)
•In the meantime, different questions could give different answer if
problem is underconstrained (as it is today)
•The statistical frontier is a moving target and as we will make progress
in understanding the first discoveries our questions will again be more
complex!
•More details now in the next two talks!
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Supplementary material after this slide
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Global NP Fit Methodology in a Nutshell
• Chose a NP model
• SUSY as show-case – so far considered models:
•
•
GUT Scale model: CMSSM, NUHM, AMSB
Soft Scale model: pMSSM
• Chose the measurements
• Considered (indirect) constraints are collider (EWK, flavour) and noncollider data (g-2, relic density)
•
It is crucial to have a consistent set of calculations for these constraints!
• Chose your preferred statistical approach to confront
model prediction Pi with measurements Mi with cov(Mi,Mi)
• “χ2 “ as an example: (M – P)T cov-1 (M – P) [+ limits]
•
Find set of Pi that minimize χ2
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
15
Constrained MSSM Analysis Pipeline
SCANNING ALGORITHM
Likelihood = 0
4 CMSSM parameters
θ = {m0, m1/2, A0, tanβ}
(fixing sign(μ) > 0)
NO
RGE
4 SM “nuisance
parameters”
Ψ={mt, mb,αS, αEM }
Data:
Gaussian likelihoods
for each of the Ψj
(j=1...4)
Non-linear
numerical
function
via SoftSusy 2.0.18
DarkSusy 4.1
MICROMEGAS 2.2
FeynHiggs 2.5.1
Hdecay 3.102
Observable
quantities
fi(θ ,Ψ)
CDM relic abundance
BR’s
EW observables
g-2
Higgs mass
sparticle spectrum
(gamma-ray, neutrino,
antimatter flux, direct
detection x-section)
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
Physically acceptable?
EWSB, no tachyons,
neutralino CDM
YES
Joint likelihood function
Data:
Gaussian likelihood
(CDM, EWO, g-2, b→sγ, ΔMBs)
other observables have
only lower/upper limits
AMSB & GMSB
0906.0957 [hep-ph]
AMSB
M0, M3/2, tanβ
“naturalness prior”
“flat prior”
GMSB
Mmess, Λ, tanβ
If Ωh2 is only used as
upper bound AMSB
seems favored.
“log prior”
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
“naturalness prior”
37
Non-Universal Higgs Models
0903.1279 [hep-ph]
“flat prior”
NUHM II
M0, M1/2, A0, tanβ, MHu2, MHd2
NUHM I
M0, M1/2, A0, tanβ, MH2
[MHu2 = MHd2]
“log prior”
0907.4468 [hep-ph] MasterCode
Also for NUHM I the frequentist
approach clearly favors
low mass SUSY!
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Closer to the Experiment
“Soft Scale Models”
A Bottom-up Approach
So far we have been far up at the GUT scale but for
the first interpretation of the hopefully soon forthcoming LHC
discoveries it will be beneficial to also consider “soft scale”
models with parameters much closer to the experimental
measurements/sensitivities (e.g. masses).
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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“Soft SUSY breaking” MSSM L
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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“Soft SUSY breaking” MSSM L
Gaugino’s and their masses M3, M2, M1
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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“Soft SUSY breaking” MSSM L
Squarks and sleptons, and their masses
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“Soft SUSY breaking” MSSM L
Tri-linear couplings A
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“Soft SUSY breaking” MSSM L
Higgs sector: 2 complex doublets (1 for u-type, 1 for d-type)
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How Much Parameters are Reasonable?
0903.1279 [hep-ph]
pMSSM:
20 NP parameter
Wow!
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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“Fitting the Soft Scale”
0903.1279 [hep-ph]
Bayesian fit
In general very strong prior
dependence – not surprising
with 20+ parameters
and only indirect constraints
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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“Fitting the Soft Scale”
Frequentist fit to 18+
parameters but now assuming
LHC input from 300/fb of data.
0907.2589 [hep-ph]
MSSM18
LHC 300/fb
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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“Fitting the Soft Scale”
Frequentist fit to 18+
parameters but now assuming
LHC input from 300/fb of data.
0907.2589 [hep-ph]
Fitting “soft scale” parameters that
are much
closer to the experimental
MSSM18
LHC 300/fb
measurements
is an interesting approach
but at least initially we need to reduce
the set of parameters.
Question: Can we define a meaningful set
Of 4 to 5 NP “soft scale” parameters?
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Global Fits and Discoveries
A few Projections
LHC Data (Discoveries) will help a lot 
If we are really luck we might see these spectacular signatures already very early at the LHC!
M(l+l-) GeV
0808.4128 [hep-ph]
0808.4128 [hep-ph]
Including a ”edge discovery”
Joint HEP-APP IOP meeting on SUSY - March0907.0594
24th 2010 [hep-ph]
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More Detailed Studies
Use MCMC for likelihood maps
Note SPS1a is close to the
minima found by global fits
(at lease MasterCode and Fittino)
SPS1a
Test running!
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Connection to Astro(particle)physics
Direct Dark Matter Searches & LHC
0809.3792 [hep-ph]
Sensitivity Plot:
WIMP(LSP) Mass vs. pSI
pSI: spin-independent dark matter WIMP
elastic scattering cross section on a free
proton.
A convenient way to illustrate direct
and indirect WIMP searches.
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
0907.4468 [hep-ph]
54
Conclusion
• Global NP Fits: Still in a tool and methodology development
phase – getting ready for the LHC
• Bayesian vs. Frequentist: Two competing schools?
• Weakly constraint NP parameter space leads to strong prior
dependences of the bayesian results.
• The frequentist fits yield consistent results favoring light mass SUSY.
• Fair question – implies the prior dependence of the bayesian fits that
the frequentist results are not conclusive too? I don’t think so!
• In any case the LHC will takes us to “statistic nirvana” (Bob Cousin)
• “Soft Scale” Fits: Botton-up approach is a very interesting
idea. Certainly closer to the experimental measurements!
•
How many parameters can we use and which ones? Can simplified
(OSET) models be a guidance?
Conclusion
• Global NP Fits @ LHC: Like in the past (e.g. EW fit, CKM
fit), global fits will be an important tool for the interpretation
of (LHC) discoveries – hence crucial for establishing the true
underlying theory of NP.
• SUSY is a show-case and we eventually will also have to consider other
NP models – BUT will we have a consistent set of calculations for the
various collider and non-collider observable in these NP models?
• I would like to thank B. Allanach, R.Trotta, A. De Roeck, F.
Ronga, the Fittion, MasterCode, Superbayes groups (and
many others) for input and very useful discussions!
• MasterCode:O. Buchmueller, R. Cavanaugh, A. De Roeck, J.R.
Ellis, H. Flacher, S. Heinemeyer, G. Isidori, K.A. Olive, F.J.
Ronga, G. Weiglein
• Fittino: P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann
• Sfitter: Remi Lafaye , Michael Rauch, Tilman Plehn, Dirk Zerwas
• S.S. AbdusSalam, B.C. Allanach, M.J. Dolan, F. Feroz, M.P.
Hobson
• Gfitter: H. Flächer, M. Goebel, J. Haller, A. Höcker, K. Mönig, J.
Stelzer
• Superbayes: L. Roszkowski, R. Ruiz de Austri, R. Trotta
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Bayesian vs. Frequentist – no math.
Bayesian approach.
This perspective on probabilities, says that a probability is a measure of a person’s degree
of belief in an event, given the information available. Thus, probabilities refer to a state of knowledge held by
an individual, rather than to the properties of a sequence of events. The use of subjective probability in
calculations of the expected value of actions is called subjective expected utility.
Frequentist approach.
They understand probability as a long-run frequency of a ‘repeatable’ event and developed
a notion of confidence intervals. Probability would be a measurable frequency of events determined from repeated
experiments. Reichenbach, Giere or Mayo have defended that approach from a philosophical point of view, referred
to by Mayo (1997) as the ‘error statistical view” (as opposed to the Bayesian or “evidential-relation view”).
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Posterior & Profile Likelihood
Slide from R. Trotta
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Constraints
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Posterior vs. Profile Likelihood
Posterior
0809.3792 [hep-ph]
Profile Likelihood
“flat prior”
“log prior”
Profile Likelihood (PL) should not depend on prior assumptions (in contrast to the Posterior pdf)
but when translating the prior dependent MCMC sampling into PLs the used sampling statistic
is often not sufficient – thus resulting in a also “prior dependent” PL result
– this should vanish with a complete sampling
Joint HEP-APP IOP meeting on SUSY - March 24th 2010
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Naturalness Prior
See e.g. 075.0487 [hep-ph]
Usually Higgs potential parameter B and μ are treated for
Mz and tanβ using:
Thus leading to the following “flat” prior assumption”
Not using the EWK symmetry breaking conditions enables uniform prior assumptions
in the more fundamental parameters B and μ: “Naturalness prior”
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Higgs Sector
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Higgs Sector
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Higgs Sector
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