Some statistical / particle physics ideas and questionslockhart/richard/banff2010/cranmer.pdf ·...
Transcript of Some statistical / particle physics ideas and questionslockhart/richard/banff2010/cranmer.pdf ·...
Kyle Cranmer (NYU) BIRS, July 13, 2010
Center for Cosmology and Particle Physics
Kyle Cranmer,New York University
Some statistical / particle physics ideas and questions
1
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Topics:‣ Follow up to the on/off problem
● Hybrid (prior predictive) method for general problems
‣ Show and tell of some complicated particle physics models
‣ Graphical models?
‣ Fisher Information Matrix/Metric and the “Asimov” data set
‣ Dealing with the look-elsewhere effect via conditioning?
2
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
This is a simplified problem that has been studied quite a bit to gain some insight into our more realistic and difficult problems‣ number counting with background uncertainty
● main measurement: observe non with s+b expected● auxiliary measurement: observe noff with expected
‣ Note: noff is used to constrain background uncertainty● In this approach “background uncertainty” is a statistical error
We learned that exact frequentist solution (construction) is formally identical to prior predictive treatment with flat prior‣ eg. choose as posterior from a flat prior and noff term
The “on/off” problem
3
ATLAS Statistics Forum
DRAFT7 May, 2010
Comments and Recommendations for Statistical Techniques
We review a collection of statistical tests used for a prototype problem, characterize their
generalizations, and provide comments on these generalizations. Where possible, concrete
recommendations are made to aid in future comparisons and combinations with ATLAS and
CMS results.
1 Preliminaries
A simple ‘prototype problem’ has been considered as useful simplification of a common HEP
situation and its coverage properties have been studied in Ref. [1] and generalized by Ref. [2].
The problem consists of a number counting analysis, where one observes non events and
expects s + b events, b is uncertain, and one either wishes to perform a significance test
against the null hypothesis s = 0 or create a confidence interval on s. Here s is considered the
parameter of interest and b is referred to as a nuisance parameter (and should be generalized
accordingly in what follows). In the setup, the background rate b is uncertain, but can
be constrained by an auxiliary or sideband measurement where one expects τb events and
measures noff events. This simple situation (often referred to as the ‘on/off’ problem) can be
expressed by the following probability density function:
P (non, noff |s, b) = Pois(non|s + b) Pois(noff |τb). (1)
Note that in this situation the sideband measurement is also modeled as a Poisson process
and the expected number of counts due to background events can be related to the main
measurement by a perfectly known ratio τ . In many cases a more accurate relation between
the sideband measurement noff and the unknown background rate b may be a Gaussian with
either an absolute or relative uncertainty ∆b. These cases were also considered in Refs. [1, 2]
and are referred to as the ‘Gaussian mean problem’.
While the prototype problem is a simplification, it has been an instructive example. The
first, and perhaps, most important lesson is that the uncertainty on the background rate bhas been cast as a well-defined statistical uncertainty instead of a vaguely-defined systematic
uncertainty. To make this point more clearly, consider that it is common practice in HEP to
describe the problem as
P (non|s) =
�db Pois(non|s + b)π(b), (2)
where π(b) is a distribution (usually Gaussian) for the uncertain parameter b, which is
then marginalized (ie. ‘smeared’, ‘randomized’, or ‘integrated out’ when creating pseudo-
experiments). But what is the nature of π(b)? The important fact which often evades serious
consideration is that π(b) is a Bayesian prior, which may or may-not be well-justified. It
often is justified by some previous measurements either based on Monte Carlo, sidebands, or
control samples. However, even in those cases one does not escape an underlying Bayesian
prior for b. The point here is not about the use of Bayesian inference, but about the clear ac-
counting of our knowledge and facilitating the ability to perform alternative statistical tests.
1
π(b)
τb
ATLAS Statistics Forum
DRAFT7 May, 2010
Comments and Recommendations for Statistical Techniques
We review a collection of statistical tests used for a prototype problem, characterize their
generalizations, and provide comments on these generalizations. Where possible, concrete
recommendations are made to aid in future comparisons and combinations with ATLAS and
CMS results.
1 Preliminaries
A simple ‘prototype problem’ has been considered as useful simplification of a common HEP
situation and its coverage properties have been studied in Ref. [1] and generalized by Ref. [2].
The problem consists of a number counting analysis, where one observes non events and
expects s + b events, b is uncertain, and one either wishes to perform a significance test
against the null hypothesis s = 0 or create a confidence interval on s. Here s is considered the
parameter of interest and b is referred to as a nuisance parameter (and should be generalized
accordingly in what follows). In the setup, the background rate b is uncertain, but can
be constrained by an auxiliary or sideband measurement where one expects τb events and
measures noff events. This simple situation (often referred to as the ‘on/off’ problem) can be
expressed by the following probability density function:
P (non, noff |s, b) = Pois(non|s + b) Pois(noff |τb). (1)
Note that in this situation the sideband measurement is also modeled as a Poisson process
and the expected number of counts due to background events can be related to the main
measurement by a perfectly known ratio τ . In many cases a more accurate relation between
the sideband measurement noff and the unknown background rate b may be a Gaussian with
either an absolute or relative uncertainty ∆b. These cases were also considered in Refs. [1, 2]
and are referred to as the ‘Gaussian mean problem’.
While the prototype problem is a simplification, it has been an instructive example. The
first, and perhaps, most important lesson is that the uncertainty on the background rate bhas been cast as a well-defined statistical uncertainty instead of a vaguely-defined systematic
uncertainty. To make this point more clearly, consider that it is common practice in HEP to
describe the problem as
P (non|s) =
�db Pois(non|s + b)π(b), (2)
where π(b) is a distribution (usually Gaussian) for the uncertain parameter b, which is
then marginalized (ie. ‘smeared’, ‘randomized’, or ‘integrated out’ when creating pseudo-
experiments). But what is the nature of π(b)? The important fact which often evades serious
consideration is that π(b) is a Bayesian prior, which may or may-not be well-justified. It
often is justified by some previous measurements either based on Monte Carlo, sidebands, or
control samples. However, even in those cases one does not escape an underlying Bayesian
prior for b. The point here is not about the use of Bayesian inference, but about the clear ac-
counting of our knowledge and facilitating the ability to perform alternative statistical tests.
1
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Goal of Bayesian-frequentist hybrid solutions is to provide a frequentist treatment of the main measurement, while eliminating nuisance parameters (deal with systematics) with an intuitive Bayesian technique.
Recommendations:‣ clearly state prior ; identify control samples or other
auxiliary measurements, then base prior on
ATLAS Statistics Forum
DRAFT7 May, 2010
Comments and Recommendations for Statistical Techniques
We review a collection of statistical tests used for a prototype problem, characterize their
generalizations, and provide comments on these generalizations. Where possible, concrete
recommendations are made to aid in future comparisons and combinations with ATLAS and
CMS results.
1 Preliminaries
A simple ‘prototype problem’ has been considered as useful simplification of a common HEP
situation and its coverage properties have been studied in Ref. [1] and generalized by Ref. [2].
The problem consists of a number counting analysis, where one observes non events and
expects s + b events, b is uncertain, and one either wishes to perform a significance test
against the null hypothesis s = 0 or create a confidence interval on s. Here s is considered the
parameter of interest and b is referred to as a nuisance parameter (and should be generalized
accordingly in what follows). In the setup, the background rate b is uncertain, but can
be constrained by an auxiliary or sideband measurement where one expects τb events and
measures noff events. This simple situation (often referred to as the ‘on/off’ problem) can be
expressed by the following probability density function:
P (non, noff |s, b) = Pois(non|s + b) Pois(noff |τb). (1)
Note that in this situation the sideband measurement is also modeled as a Poisson process
and the expected number of counts due to background events can be related to the main
measurement by a perfectly known ratio τ . In many cases a more accurate relation between
the sideband measurement noff and the unknown background rate b may be a Gaussian with
either an absolute or relative uncertainty ∆b. These cases were also considered in Refs. [1, 2]
and are referred to as the ‘Gaussian mean problem’.
While the prototype problem is a simplification, it has been an instructive example. The
first, and perhaps, most important lesson is that the uncertainty on the background rate bhas been cast as a well-defined statistical uncertainty instead of a vaguely-defined systematic
uncertainty. To make this point more clearly, consider that it is common practice in HEP to
describe the problem as
P (non|s) =
�db Pois(non|s + b)π(b), (2)
where π(b) is a distribution (usually Gaussian) for the uncertain parameter b, which is
then marginalized (ie. ‘smeared’, ‘randomized’, or ‘integrated out’ when creating pseudo-
experiments). But what is the nature of π(b)? The important fact which often evades serious
consideration is that π(b) is a Bayesian prior, which may or may-not be well-justified. It
often is justified by some previous measurements either based on Monte Carlo, sidebands, or
control samples. However, even in those cases one does not escape an underlying Bayesian
prior for b. The point here is not about the use of Bayesian inference, but about the clear ac-
counting of our knowledge and facilitating the ability to perform alternative statistical tests.
1
Bayesian-Frequentist Hybrid Solutions
4
η(b)If we were actually in a case described by the ‘on/off’ problem, then it would be better tothink of π(b) as the posterior resulting from the sideband measurement
π(b) = P (b|noff) =P (noff |b)η(b)�dbP (noff |b)η(b)
. (3)
By doing this it is clear that the term P (noff |b) is an objective probability density that canbe used in a frequentist context and that η(b) is the original Bayesian prior assigned to b.
Recommendation: Where possible, one should express uncertainty on a parameter asstatistical (eg. random) process (ie. Pois(noff |τb) in Eq. 1).
Recommendation: When using Bayesian techniques, one should explicitly express andseparate the prior from the objective part of the probability density function (as in Eq. 3).
Now let us consider some specific methods for addressing the on/off problem and theirgeneralizations.
2 The frequentist solution: ZBi
The goal for a frequentist solution to this problem is based on the notion of coverage (orType I error). One considers there to be some unknown true values for the parameters s, band attempts to construct a statistical test that will not incorrectly reject the true valuesabove some specified rate α.
A frequentist solution to the on/off problem, referred to as ZBi in Refs. [1, 2], is based onre-writing Eq. 1 into a different form and using the standard frequentist binomial parametertest, which dates back to the first construction of confidence intervals for a binomial parameterby Clopper and Pearson in 1934 [3]. This does not lead to an obvious generalization for morecomplex problems.
The general solution to this problem, which provides coverage “by construction” is theNeyman Construction. However, the Neyman Construction is not uniquely determined; onemust also specify:
• the test statistic T (non, noff ; s, b), which depends on data and parameters
• a well-defined ensemble that defines the sampling distribution of T
• the limits of integration for the sampling distribution of T
• parameter points to scan (including the values of any nuisance parameters)
• how the final confidence intervals in the parameter of interest are established
The Feldman-Cousins technique is a well-specified Neyman Construction when there areno nuisance parameters [6]: the test statistic is the likelihood ratio T (non; s) = L(s)/L(sbest),the limits of integration are one-sided, there is no special conditioning done to the ensemble,and there are no nuisance parameters to complicate the scanning of the parameter points orthe construction of the final intervals.
The original Feldman-Cousins paper did not specify a technique for dealing with nuisanceparameters, but several generalization have been proposed. The bulk of the variations comefrom the choice of the test statistic to use.
2
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Generalizing the Hybrid approachIn RooStats we are providing several techniques given a common specification of the problem that relies on:‣ the joint model ‣ a Bayesian prior ‣ and some data
The question is “how do we generalize the Hybrid (prior predictive) approach” given this information
5
P (x, y|s, b, τ)η(s, b)
(x0, y0)
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Generalizing the Hybrid approachStart with “on/off” example‣ the joint model ‣ a Bayesian prior
How do we identify the “off” part of the model‣ was an average model for non, so use largest factor
independent of non, or ‣ think find largest parameter independent of parameter of
interest
6
P (x, y|s, b, τ)
η(s, b)
ATLAS Statistics Forum
DRAFT7 May, 2010
Comments and Recommendations for Statistical Techniques
We review a collection of statistical tests used for a prototype problem, characterize their
generalizations, and provide comments on these generalizations. Where possible, concrete
recommendations are made to aid in future comparisons and combinations with ATLAS and
CMS results.
1 Preliminaries
A simple ‘prototype problem’ has been considered as useful simplification of a common HEP
situation and its coverage properties have been studied in Ref. [1] and generalized by Ref. [2].
The problem consists of a number counting analysis, where one observes non events and
expects s + b events, b is uncertain, and one either wishes to perform a significance test
against the null hypothesis s = 0 or create a confidence interval on s. Here s is considered the
parameter of interest and b is referred to as a nuisance parameter (and should be generalized
accordingly in what follows). In the setup, the background rate b is uncertain, but can
be constrained by an auxiliary or sideband measurement where one expects τb events and
measures noff events. This simple situation (often referred to as the ‘on/off’ problem) can be
expressed by the following probability density function:
P (non, noff |s, b) = Pois(non|s + b) Pois(noff |τb). (1)
Note that in this situation the sideband measurement is also modeled as a Poisson process
and the expected number of counts due to background events can be related to the main
measurement by a perfectly known ratio τ . In many cases a more accurate relation between
the sideband measurement noff and the unknown background rate b may be a Gaussian with
either an absolute or relative uncertainty ∆b. These cases were also considered in Refs. [1, 2]
and are referred to as the ‘Gaussian mean problem’.
While the prototype problem is a simplification, it has been an instructive example. The
first, and perhaps, most important lesson is that the uncertainty on the background rate bhas been cast as a well-defined statistical uncertainty instead of a vaguely-defined systematic
uncertainty. To make this point more clearly, consider that it is common practice in HEP to
describe the problem as
P (non|s) =
�db Pois(non|s + b)π(b), (2)
where π(b) is a distribution (usually Gaussian) for the uncertain parameter b, which is
then marginalized (ie. ‘smeared’, ‘randomized’, or ‘integrated out’ when creating pseudo-
experiments). But what is the nature of π(b)? The important fact which often evades serious
consideration is that π(b) is a Bayesian prior, which may or may-not be well-justified. It
often is justified by some previous measurements either based on Monte Carlo, sidebands, or
control samples. However, even in those cases one does not escape an underlying Bayesian
prior for b. The point here is not about the use of Bayesian inference, but about the clear ac-
counting of our knowledge and facilitating the ability to perform alternative statistical tests.
1
If we were actually in a case described by the ‘on/off’ problem, then it would be better tothink of π(b) as the posterior resulting from the sideband measurement
π(b) = P (b|noff) =P (noff |b)η(b)�dbP (noff |b)η(b)
. (3)
By doing this it is clear that the term P (noff |b) is an objective probability density that canbe used in a frequentist context and that η(b) is the original Bayesian prior assigned to b.
Recommendation: Where possible, one should express uncertainty on a parameter asstatistical (eg. random) process (ie. Pois(noff |τb) in Eq. 1).
Recommendation: When using Bayesian techniques, one should explicitly express andseparate the prior from the objective part of the probability density function (as in Eq. 3).
Now let us consider some specific methods for addressing the on/off problem and theirgeneralizations.
2 The frequentist solution: ZBi
The goal for a frequentist solution to this problem is based on the notion of coverage (orType I error). One considers there to be some unknown true values for the parameters s, band attempts to construct a statistical test that will not incorrectly reject the true valuesabove some specified rate α.
A frequentist solution to the on/off problem, referred to as ZBi in Refs. [1, 2], is based onre-writing Eq. 1 into a different form and using the standard frequentist binomial parametertest, which dates back to the first construction of confidence intervals for a binomial parameterby Clopper and Pearson in 1934 [3]. This does not lead to an obvious generalization for morecomplex problems.
The general solution to this problem, which provides coverage “by construction” is theNeyman Construction. However, the Neyman Construction is not uniquely determined; onemust also specify:
• the test statistic T (non, noff ; s, b), which depends on data and parameters
• a well-defined ensemble that defines the sampling distribution of T
• the limits of integration for the sampling distribution of T
• parameter points to scan (including the values of any nuisance parameters)
• how the final confidence intervals in the parameter of interest are established
The Feldman-Cousins technique is a well-specified Neyman Construction when there areno nuisance parameters [6]: the test statistic is the likelihood ratio T (non; s) = L(s)/L(sbest),the limits of integration are one-sided, there is no special conditioning done to the ensemble,and there are no nuisance parameters to complicate the scanning of the parameter points orthe construction of the final intervals.
The original Feldman-Cousins paper did not specify a technique for dealing with nuisanceparameters, but several generalization have been proposed. The bulk of the variations comefrom the choice of the test statistic to use.
2
ATLAS Statistics Forum
DRAFT7 May, 2010
Comments and Recommendations for Statistical Techniques
We review a collection of statistical tests used for a prototype problem, characterize their
generalizations, and provide comments on these generalizations. Where possible, concrete
recommendations are made to aid in future comparisons and combinations with ATLAS and
CMS results.
1 Preliminaries
A simple ‘prototype problem’ has been considered as useful simplification of a common HEP
situation and its coverage properties have been studied in Ref. [1] and generalized by Ref. [2].
The problem consists of a number counting analysis, where one observes non events and
expects s + b events, b is uncertain, and one either wishes to perform a significance test
against the null hypothesis s = 0 or create a confidence interval on s. Here s is considered the
parameter of interest and b is referred to as a nuisance parameter (and should be generalized
accordingly in what follows). In the setup, the background rate b is uncertain, but can
be constrained by an auxiliary or sideband measurement where one expects τb events and
measures noff events. This simple situation (often referred to as the ‘on/off’ problem) can be
expressed by the following probability density function:
P (non, noff |s, b) = Pois(non|s + b) Pois(noff |τb). (1)
Note that in this situation the sideband measurement is also modeled as a Poisson process
and the expected number of counts due to background events can be related to the main
measurement by a perfectly known ratio τ . In many cases a more accurate relation between
the sideband measurement noff and the unknown background rate b may be a Gaussian with
either an absolute or relative uncertainty ∆b. These cases were also considered in Refs. [1, 2]
and are referred to as the ‘Gaussian mean problem’.
While the prototype problem is a simplification, it has been an instructive example. The
first, and perhaps, most important lesson is that the uncertainty on the background rate bhas been cast as a well-defined statistical uncertainty instead of a vaguely-defined systematic
uncertainty. To make this point more clearly, consider that it is common practice in HEP to
describe the problem as
P (non|s) =
�db Pois(non|s + b)π(b), (2)
where π(b) is a distribution (usually Gaussian) for the uncertain parameter b, which is
then marginalized (ie. ‘smeared’, ‘randomized’, or ‘integrated out’ when creating pseudo-
experiments). But what is the nature of π(b)? The important fact which often evades serious
consideration is that π(b) is a Bayesian prior, which may or may-not be well-justified. It
often is justified by some previous measurements either based on Monte Carlo, sidebands, or
control samples. However, even in those cases one does not escape an underlying Bayesian
prior for b. The point here is not about the use of Bayesian inference, but about the clear ac-
counting of our knowledge and facilitating the ability to perform alternative statistical tests.
1
P
Pon
non s b
Poff
noff !
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Generalizing the Hybrid approachThe two approaches are not equivalent for joint models like this:‣ was an average model for non, so use largest factor independent of non, or ‣ think find largest parameter independent of parameter of interest
And then there is the question of the prior... what if doesn’t factorize‣ marginal over s will have some residual prior dependence on s
For numerical reasons, we would like to have a table of replacements:
7
P
Pon
non s b
Poff
noff !
P
Pon
non s b
Poff
noff !
η(s, b)
PDF Prior PosteriorGaussian uniform GaussianPoisson uniform GammaLog-normal reference Log-Normal
π
η
b
Poff
noff !b
Γ
noff !
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010 8
The RooFit/RooStats workspace
RooFit’s Workspace now provides the ability to save in a ROOT file the full probability model, any priors you might need, and the minimal data necessary to reproduce likelihood function.
Need this for combinations, exciting potential for publishing results.
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
ATLAS H->γγ
9
RooSimultaneous
simPdf_reparametrized_hgg
RooSuperCategory
fitCat
RooCategory
etaCat
RooCategory
convCat
RooCategory
jetCat
RooAddPdf
sumPdf_{etaGood;noConv;noJets}_reparametrized_hgg
RooProdPdf
Background_{etaGood;noConv;noJets}
Hfitter::MggBkgPdf
Background_mgg_{etaGood;noConv;noJets}
RooRealVar
offset
RooRealVar
xi_noJets
RooRealVar
mgg
RooAddPdf
Background_cosThStar_{etaGood;noConv;noJets}
RooGaussian
Background_bump_cosThStar_2
RooRealVar
bkg_csth02
RooRealVar
bkg_csthSigma2
RooRealVar
cosThStar
RooRealVar
bkg_csthRelNorm2
RooGaussian
Background_bump_cosThStar_1_{etaGood;noConv;noJets}
RooRealVar
bkg_csth01_{etaGood;noJets}
RooRealVar
bkg_csthSigma1_{etaGood;noJets}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaGood;noConv;noJets}
RooRealVar
bkg_csthCoef3
RooRealVar
bkg_csthCoef5
RooRealVar
bkg_csthPower_{etaGood;noJets}
RooRealVar
bkg_csthCoef1_{etaGood;noJets}
RooRealVar
bkg_csthCoef2_{etaGood;noJets}
RooRealVar
bkg_csthCoef4_{etaGood;noJets}
RooRealVar
bkg_csthCoef6_{etaGood;noJets}
RooRealVar
bkg_csthRelNorm1_{etaGood;noJets}
RooAddPdf
Background_pT_{etaGood;noConv;noJets}
Hfitter::HftPolyExp
Background_pT_3
RooRealVar
bkg_ptM0
RooRealVar
bkg_ptPow3
RooRealVar
bkg_ptExp3
RooRealVar
pT
RooRealVar
bkg_ptTail2Norm
RooGaussian
Background_pT_TC
RooRealVar
bkg_ptTCS
RooRealVar
bkg_ptTailTCNorm
Hfitter::HftPolyExp
Background_pT_2_{etaGood;noConv;noJets}
RooRealVar
bkg_ptPow2_noJets
RooRealVar
bkg_ptExp2_noJets
Hfitter::HftPolyExp
Background_pT_1_{etaGood;noConv;noJets}
RooRealVar
bkg_ptPow1_noJets
RooRealVar
bkg_ptExp1_noJets
RooRealVar
bkg_ptTail1Norm_noJets
RooProdPdf
Signal_{etaGood;noConv;noJets}
RooAddPdf
Signal_mgg_{etaGood;noConv;noJets}
RooGaussian
Signal_mgg_Tail
RooRealVar
mTail
RooRealVar
sigTail
RooCBShape
Signal_mgg_Peak_{etaGood;noConv;noJets}
RooFormulaVar
mHiggsFormula_{etaGood;noConv;noJets}
RooRealVar
mHiggs
RooRealVar
dmHiggs_{etaGood;noConv}
RooRealVar
mRes_{etaGood;noConv}
RooRealVar
tailAlpha_{etaGood;noConv}
RooRealVar
tailN_{etaGood;noConv}
RooRealVar
mggRelNorm_{etaGood;noConv}
RooAddPdf
Signal_cosThStar_{etaGood;noConv;noJets}
RooGaussian
Signalcsthstr_1_{etaGood;noConv;noJets}
RooRealVar
sig_csthstrSigma_1
RooRealVar
sig_csthstr0_1_{etaGood;noJets}
RooGaussian
Signalcsthstr_2_{etaGood;noConv;noJets}
RooRealVar
sig_csthstrSigma_2
RooRealVar
sig_csthstr0_2_{etaGood;noJets}
RooRealVar
sig_csthstrRelNorm_1_{etaGood;noJets}
RooAddPdf
Signal_pT_{etaGood;noConv;noJets}
RooBifurGauss
SignalpT_4
RooRealVar
sig_pt0_4
RooRealVar
sig_ptSigmaL_4
RooRealVar
sig_ptSigmaR_4
RooRealVar
sig_ptRelNorm_4
RooBifurGauss
SignalpT_1_{etaGood;noConv;noJets}
RooRealVar
sig_pt0_1_noJets
RooRealVar
sig_ptSigmaL_1_noJets
RooRealVar
sig_ptSigmaR_1_noJets
RooGaussian
SignalpT_2_{etaGood;noConv;noJets}
RooRealVar
sig_pt0_2_noJets
RooRealVar
sig_ptSigma_2_noJets
RooGaussian
SignalpT_3_{etaGood;noConv;noJets}
RooRealVar
sig_pt0_3_noJets
RooRealVar
sig_ptSigma_3_noJets
RooRealVar
sig_ptRelNorm_1_noJets
RooRealVar
sig_ptRelNorm_2_noJets
RooRealVar
n_Background_{etaGood;noConv;noJets}
RooFormulaVar
nCat_Signal_{etaGood;noConv;noJets}_reparametrized_hgg
RooRealVar
f_Signal_{etaGood;noConv;noJets}
RooProduct
new_n_Signal
RooRealVar
mu
RooRealVar
n_Signal
RooAddPdf
sumPdf_{etaMed;noConv;noJets}_reparametrized_hgg
RooProdPdf
Background_{etaMed;noConv;noJets}
Hfitter::MggBkgPdf
Background_mgg_{etaMed;noConv;noJets}
RooAddPdf
Background_cosThStar_{etaMed;noConv;noJets}
RooGaussian
Background_bump_cosThStar_1_{etaMed;noConv;noJets}
RooRealVar
bkg_csth01_{etaMed;noJets}
RooRealVar
bkg_csthSigma1_{etaMed;noJets}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaMed;noConv;noJets}
RooRealVar
bkg_csthPower_{etaMed;noJets}
RooRealVar
bkg_csthCoef1_{etaMed;noJets}
RooRealVar
bkg_csthCoef2_{etaMed;noJets}
RooRealVar
bkg_csthCoef4_{etaMed;noJets}
RooRealVar
bkg_csthCoef6_{etaMed;noJets}
RooRealVar
bkg_csthRelNorm1_{etaMed;noJets}
RooAddPdf
Background_pT_{etaMed;noConv;noJets}
Hfitter::HftPolyExp
Background_pT_2_{etaMed;noConv;noJets}
Hfitter::HftPolyExp
Background_pT_1_{etaMed;noConv;noJets}
RooProdPdf
Signal_{etaMed;noConv;noJets}
RooAddPdf
Signal_mgg_{etaMed;noConv;noJets}
RooCBShape
Signal_mgg_Peak_{etaMed;noConv;noJets}
RooFormulaVar
mHiggsFormula_{etaMed;noConv;noJets}
RooRealVar
dmHiggs_{etaMed;noConv}
RooRealVar
mRes_{etaMed;noConv}
RooRealVar
tailAlpha_{etaMed;noConv}
RooRealVar
tailN_{etaMed;noConv}
RooRealVar
mggRelNorm_{etaMed;noConv}
RooAddPdf
Signal_cosThStar_{etaMed;noConv;noJets}
RooGaussian
Signalcsthstr_1_{etaMed;noConv;noJets}
RooRealVar
sig_csthstr0_1_{etaMed;noJets}
RooGaussian
Signalcsthstr_2_{etaMed;noConv;noJets}
RooRealVar
sig_csthstr0_2_{etaMed;noJets}
RooRealVar
sig_csthstrRelNorm_1_{etaMed;noJets}
RooAddPdf
Signal_pT_{etaMed;noConv;noJets}
RooBifurGauss
SignalpT_1_{etaMed;noConv;noJets}
RooGaussian
SignalpT_2_{etaMed;noConv;noJets}
RooGaussian
SignalpT_3_{etaMed;noConv;noJets}
RooRealVar
n_Background_{etaMed;noConv;noJets}
RooFormulaVar
nCat_Signal_{etaMed;noConv;noJets}_reparametrized_hgg
RooRealVar
f_Signal_{etaMed;noConv;noJets}
RooAddPdf
sumPdf_{etaBad;noConv;noJets}_reparametrized_hgg
RooProdPdf
Background_{etaBad;noConv;noJets}
Hfitter::MggBkgPdf
Background_mgg_{etaBad;noConv;noJets}
RooAddPdf
Background_cosThStar_{etaBad;noConv;noJets}
RooGaussian
Background_bump_cosThStar_1_{etaBad;noConv;noJets}
RooRealVar
bkg_csth01_{etaBad;noJets}
RooRealVar
bkg_csthSigma1_{etaBad;noJets}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaBad;noConv;noJets}
RooRealVar
bkg_csthPower_{etaBad;noJets}
RooRealVar
bkg_csthCoef1_{etaBad;noJets}
RooRealVar
bkg_csthCoef2_{etaBad;noJets}
RooRealVar
bkg_csthCoef4_{etaBad;noJets}
RooRealVar
bkg_csthCoef6_{etaBad;noJets}
RooRealVar
bkg_csthRelNorm1_{etaBad;noJets}
RooAddPdf
Background_pT_{etaBad;noConv;noJets}
Hfitter::HftPolyExp
Background_pT_2_{etaBad;noConv;noJets}
Hfitter::HftPolyExp
Background_pT_1_{etaBad;noConv;noJets}
RooProdPdf
Signal_{etaBad;noConv;noJets}
RooAddPdf
Signal_mgg_{etaBad;noConv;noJets}
RooCBShape
Signal_mgg_Peak_{etaBad;noConv;noJets}
RooFormulaVar
mHiggsFormula_{etaBad;noConv;noJets}
RooRealVar
dmHiggs_{etaBad;noConv}
RooRealVar
mRes_{etaBad;noConv}
RooRealVar
tailAlpha_{etaBad;noConv}
RooRealVar
tailN_{etaBad;noConv}
RooRealVar
mggRelNorm_{etaBad;noConv}
RooAddPdf
Signal_cosThStar_{etaBad;noConv;noJets}
RooGaussian
Signalcsthstr_1_{etaBad;noConv;noJets}
RooRealVar
sig_csthstr0_1_{etaBad;noJets}
RooGaussian
Signalcsthstr_2_{etaBad;noConv;noJets}
RooRealVar
sig_csthstr0_2_{etaBad;noJets}
RooRealVar
sig_csthstrRelNorm_1_{etaBad;noJets}
RooAddPdf
Signal_pT_{etaBad;noConv;noJets}
RooBifurGauss
SignalpT_1_{etaBad;noConv;noJets}
RooGaussian
SignalpT_2_{etaBad;noConv;noJets}
RooGaussian
SignalpT_3_{etaBad;noConv;noJets}
RooRealVar
n_Background_{etaBad;noConv;noJets}
RooFormulaVar
nCat_Signal_{etaBad;noConv;noJets}_reparametrized_hgg
RooRealVar
f_Signal_{etaBad;noConv;noJets}
RooAddPdf
sumPdf_{etaGood;Conv;noJets}_reparametrized_hgg
RooProdPdf
Background_{etaGood;Conv;noJets}
Hfitter::MggBkgPdf
Background_mgg_{etaGood;Conv;noJets}
RooAddPdf
Background_cosThStar_{etaGood;Conv;noJets}
RooGaussian
Background_bump_cosThStar_1_{etaGood;Conv;noJets}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaGood;Conv;noJets}
RooAddPdf
Background_pT_{etaGood;Conv;noJets}
Hfitter::HftPolyExp
Background_pT_2_{etaGood;Conv;noJets}
Hfitter::HftPolyExp
Background_pT_1_{etaGood;Conv;noJets}
RooProdPdf
Signal_{etaGood;Conv;noJets}
RooAddPdf
Signal_mgg_{etaGood;Conv;noJets}
RooCBShape
Signal_mgg_Peak_{etaGood;Conv;noJets}
RooFormulaVar
mHiggsFormula_{etaGood;Conv;noJets}
RooRealVar
dmHiggs_{etaGood;Conv}
RooRealVar
mRes_{etaGood;Conv}
RooRealVar
tailAlpha_{etaGood;Conv}
RooRealVar
tailN_{etaGood;Conv}
RooRealVar
mggRelNorm_{etaGood;Conv}
RooAddPdf
Signal_cosThStar_{etaGood;Conv;noJets}
RooGaussian
Signalcsthstr_1_{etaGood;Conv;noJets}
RooGaussian
Signalcsthstr_2_{etaGood;Conv;noJets}
RooAddPdf
Signal_pT_{etaGood;Conv;noJets}
RooBifurGauss
SignalpT_1_{etaGood;Conv;noJets}
RooGaussian
SignalpT_2_{etaGood;Conv;noJets}
RooGaussian
SignalpT_3_{etaGood;Conv;noJets}
RooRealVar
n_Background_{etaGood;Conv;noJets}
RooFormulaVar
nCat_Signal_{etaGood;Conv;noJets}_reparametrized_hgg
RooRealVar
f_Signal_{etaGood;Conv;noJets}
RooAddPdf
sumPdf_{etaMed;Conv;noJets}_reparametrized_hgg
RooProdPdf
Background_{etaMed;Conv;noJets}
Hfitter::MggBkgPdf
Background_mgg_{etaMed;Conv;noJets}
RooAddPdf
Background_cosThStar_{etaMed;Conv;noJets}
RooGaussian
Background_bump_cosThStar_1_{etaMed;Conv;noJets}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaMed;Conv;noJets}
RooAddPdf
Background_pT_{etaMed;Conv;noJets}
Hfitter::HftPolyExp
Background_pT_2_{etaMed;Conv;noJets}
Hfitter::HftPolyExp
Background_pT_1_{etaMed;Conv;noJets}
RooProdPdf
Signal_{etaMed;Conv;noJets}
RooAddPdf
Signal_mgg_{etaMed;Conv;noJets}
RooCBShape
Signal_mgg_Peak_{etaMed;Conv;noJets}
RooFormulaVar
mHiggsFormula_{etaMed;Conv;noJets}
RooRealVar
dmHiggs_{etaMed;Conv}
RooRealVar
mRes_{etaMed;Conv}
RooRealVar
tailAlpha_{etaMed;Conv}
RooRealVar
tailN_{etaMed;Conv}
RooRealVar
mggRelNorm_{etaMed;Conv}
RooAddPdf
Signal_cosThStar_{etaMed;Conv;noJets}
RooGaussian
Signalcsthstr_1_{etaMed;Conv;noJets}
RooGaussian
Signalcsthstr_2_{etaMed;Conv;noJets}
RooAddPdf
Signal_pT_{etaMed;Conv;noJets}
RooBifurGauss
SignalpT_1_{etaMed;Conv;noJets}
RooGaussian
SignalpT_2_{etaMed;Conv;noJets}
RooGaussian
SignalpT_3_{etaMed;Conv;noJets}
RooRealVar
n_Background_{etaMed;Conv;noJets}
RooFormulaVar
nCat_Signal_{etaMed;Conv;noJets}_reparametrized_hgg
RooRealVar
f_Signal_{etaMed;Conv;noJets}
RooAddPdf
sumPdf_{etaBad;Conv;noJets}_reparametrized_hgg
RooProdPdf
Background_{etaBad;Conv;noJets}
Hfitter::MggBkgPdf
Background_mgg_{etaBad;Conv;noJets}
RooAddPdf
Background_cosThStar_{etaBad;Conv;noJets}
RooGaussian
Background_bump_cosThStar_1_{etaBad;Conv;noJets}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaBad;Conv;noJets}
RooAddPdf
Background_pT_{etaBad;Conv;noJets}
Hfitter::HftPolyExp
Background_pT_2_{etaBad;Conv;noJets}
Hfitter::HftPolyExp
Background_pT_1_{etaBad;Conv;noJets}
RooProdPdf
Signal_{etaBad;Conv;noJets}
RooAddPdf
Signal_mgg_{etaBad;Conv;noJets}
RooCBShape
Signal_mgg_Peak_{etaBad;Conv;noJets}
RooFormulaVar
mHiggsFormula_{etaBad;Conv;noJets}
RooRealVar
dmHiggs_{etaBad;Conv}
RooRealVar
mRes_{etaBad;Conv}
RooRealVar
tailAlpha_{etaBad;Conv}
RooRealVar
tailN_{etaBad;Conv}
RooRealVar
mggRelNorm_{etaBad;Conv}
RooAddPdf
Signal_cosThStar_{etaBad;Conv;noJets}
RooGaussian
Signalcsthstr_1_{etaBad;Conv;noJets}
RooGaussian
Signalcsthstr_2_{etaBad;Conv;noJets}
RooAddPdf
Signal_pT_{etaBad;Conv;noJets}
RooBifurGauss
SignalpT_1_{etaBad;Conv;noJets}
RooGaussian
SignalpT_2_{etaBad;Conv;noJets}
RooGaussian
SignalpT_3_{etaBad;Conv;noJets}
RooRealVar
n_Background_{etaBad;Conv;noJets}
RooFormulaVar
nCat_Signal_{etaBad;Conv;noJets}_reparametrized_hgg
RooRealVar
f_Signal_{etaBad;Conv;noJets}
RooAddPdf
sumPdf_{etaGood;noConv;jet}_reparametrized_hgg
RooProdPdf
Background_{etaGood;noConv;jet}
Hfitter::MggBkgPdf
Background_mgg_{etaGood;noConv;jet}
RooRealVar
xi_jet
RooAddPdf
Background_cosThStar_{etaGood;noConv;jet}
RooGaussian
Background_bump_cosThStar_1_{etaGood;noConv;jet}
RooRealVar
bkg_csth01_{etaGood;jet}
RooRealVar
bkg_csthSigma1_{etaGood;jet}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaGood;noConv;jet}
RooRealVar
bkg_csthRelNorm1_{etaGood;jet}
RooAddPdf
Background_pT_{etaGood;noConv;jet}
Hfitter::HftPolyExp
Background_pT_2_{etaGood;noConv;jet}
Hfitter::HftPolyExp
Background_pT_1_{etaGood;noConv;jet}
RooRealVar
bkg_ptPow1_jet
RooRealVar
bkg_ptExp1_jet
RooProdPdf
Signal_{etaGood;noConv;jet}
RooAddPdf
Signal_mgg_{etaGood;noConv;jet}
RooCBShape
Signal_mgg_Peak_{etaGood;noConv;jet}
RooFormulaVar
mHiggsFormula_{etaGood;noConv;jet}
RooAddPdf
Signal_cosThStar_{etaGood;noConv;jet}
RooGaussian
Signalcsthstr_1_{etaGood;noConv;jet}
RooGaussian
Signalcsthstr_2_{etaGood;noConv;jet}
RooAddPdf
Signal_pT_{etaGood;noConv;jet}
RooBifurGauss
SignalpT_1_{etaGood;noConv;jet}
RooRealVar
sig_pt0_1_jet
RooRealVar
sig_ptSigmaL_1_jet
RooRealVar
sig_ptSigmaR_1_jet
RooGaussian
SignalpT_2_{etaGood;noConv;jet}
RooRealVar
sig_pt0_2_jet
RooRealVar
sig_ptSigma_2_jet
RooGaussian
SignalpT_3_{etaGood;noConv;jet}
RooRealVar
sig_pt0_3_jet
RooRealVar
sig_ptSigma_3_jet
RooRealVar
sig_ptRelNorm_1_jet
RooRealVar
sig_ptRelNorm_2_jet
RooRealVar
n_Background_{etaGood;noConv;jet}
RooFormulaVar
nCat_Signal_{etaGood;noConv;jet}_reparametrized_hgg
RooRealVar
f_Signal_{etaGood;noConv;jet}
RooAddPdf
sumPdf_{etaMed;noConv;jet}_reparametrized_hgg
RooProdPdf
Background_{etaMed;noConv;jet}
Hfitter::MggBkgPdf
Background_mgg_{etaMed;noConv;jet}
RooAddPdf
Background_cosThStar_{etaMed;noConv;jet}
RooGaussian
Background_bump_cosThStar_1_{etaMed;noConv;jet}
RooRealVar
bkg_csth01_{etaMed;jet}
RooRealVar
bkg_csthSigma1_{etaMed;jet}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaMed;noConv;jet}
RooRealVar
bkg_csthPower_{etaMed;jet}
RooRealVar
bkg_csthCoef1_{etaMed;jet}
RooRealVar
bkg_csthCoef2_{etaMed;jet}
RooRealVar
bkg_csthCoef4_{etaMed;jet}
RooRealVar
bkg_csthCoef6_{etaMed;jet}
RooRealVar
bkg_csthRelNorm1_{etaMed;jet}
RooAddPdf
Background_pT_{etaMed;noConv;jet}
Hfitter::HftPolyExp
Background_pT_2_{etaMed;noConv;jet}
Hfitter::HftPolyExp
Background_pT_1_{etaMed;noConv;jet}
RooProdPdf
Signal_{etaMed;noConv;jet}
RooAddPdf
Signal_mgg_{etaMed;noConv;jet}
RooCBShape
Signal_mgg_Peak_{etaMed;noConv;jet}
RooFormulaVar
mHiggsFormula_{etaMed;noConv;jet}
RooAddPdf
Signal_cosThStar_{etaMed;noConv;jet}
RooGaussian
Signalcsthstr_1_{etaMed;noConv;jet}
RooRealVar
sig_csthstr0_1_{etaMed;jet}
RooGaussian
Signalcsthstr_2_{etaMed;noConv;jet}
RooRealVar
sig_csthstr0_2_{etaMed;jet}
RooRealVar
sig_csthstrRelNorm_1_{etaMed;jet}
RooAddPdf
Signal_pT_{etaMed;noConv;jet}
RooBifurGauss
SignalpT_1_{etaMed;noConv;jet}
RooGaussian
SignalpT_2_{etaMed;noConv;jet}
RooGaussian
SignalpT_3_{etaMed;noConv;jet}
RooRealVar
n_Background_{etaMed;noConv;jet}
RooFormulaVar
nCat_Signal_{etaMed;noConv;jet}_reparametrized_hgg
RooRealVar
f_Signal_{etaMed;noConv;jet}
RooAddPdf
sumPdf_{etaBad;noConv;jet}_reparametrized_hgg
RooProdPdf
Background_{etaBad;noConv;jet}
Hfitter::MggBkgPdf
Background_mgg_{etaBad;noConv;jet}
RooAddPdf
Background_cosThStar_{etaBad;noConv;jet}
RooGaussian
Background_bump_cosThStar_1_{etaBad;noConv;jet}
RooRealVar
bkg_csth01_{etaBad;jet}
RooRealVar
bkg_csthSigma1_{etaBad;jet}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaBad;noConv;jet}
RooRealVar
bkg_csthPower_{etaBad;jet}
RooRealVar
bkg_csthCoef1_{etaBad;jet}
RooRealVar
bkg_csthCoef2_{etaBad;jet}
RooRealVar
bkg_csthCoef4_{etaBad;jet}
RooRealVar
bkg_csthCoef6_{etaBad;jet}
RooRealVar
bkg_csthRelNorm1_{etaBad;jet}
RooAddPdf
Background_pT_{etaBad;noConv;jet}
Hfitter::HftPolyExp
Background_pT_2_{etaBad;noConv;jet}
Hfitter::HftPolyExp
Background_pT_1_{etaBad;noConv;jet}
RooProdPdf
Signal_{etaBad;noConv;jet}
RooAddPdf
Signal_mgg_{etaBad;noConv;jet}
RooCBShape
Signal_mgg_Peak_{etaBad;noConv;jet}
RooFormulaVar
mHiggsFormula_{etaBad;noConv;jet}
RooAddPdf
Signal_cosThStar_{etaBad;noConv;jet}
RooGaussian
Signalcsthstr_1_{etaBad;noConv;jet}
RooRealVar
sig_csthstr0_1_{etaBad;jet}
RooGaussian
Signalcsthstr_2_{etaBad;noConv;jet}
RooRealVar
sig_csthstr0_2_{etaBad;jet}
RooRealVar
sig_csthstrRelNorm_1_{etaBad;jet}
RooAddPdf
Signal_pT_{etaBad;noConv;jet}
RooBifurGauss
SignalpT_1_{etaBad;noConv;jet}
RooGaussian
SignalpT_2_{etaBad;noConv;jet}
RooGaussian
SignalpT_3_{etaBad;noConv;jet}
RooRealVar
n_Background_{etaBad;noConv;jet}
RooFormulaVar
nCat_Signal_{etaBad;noConv;jet}_reparametrized_hgg
RooRealVar
f_Signal_{etaBad;noConv;jet}
RooAddPdf
sumPdf_{etaMed;Conv;jet}_reparametrized_hgg
RooProdPdf
Background_{etaMed;Conv;jet}
Hfitter::MggBkgPdf
Background_mgg_{etaMed;Conv;jet}
RooAddPdf
Background_cosThStar_{etaMed;Conv;jet}
RooGaussian
Background_bump_cosThStar_1_{etaMed;Conv;jet}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaMed;Conv;jet}
RooAddPdf
Background_pT_{etaMed;Conv;jet}
Hfitter::HftPolyExp
Background_pT_2_{etaMed;Conv;jet}
Hfitter::HftPolyExp
Background_pT_1_{etaMed;Conv;jet}
RooProdPdf
Signal_{etaMed;Conv;jet}
RooAddPdf
Signal_mgg_{etaMed;Conv;jet}
RooCBShape
Signal_mgg_Peak_{etaMed;Conv;jet}
RooFormulaVar
mHiggsFormula_{etaMed;Conv;jet}
RooAddPdf
Signal_cosThStar_{etaMed;Conv;jet}
RooGaussian
Signalcsthstr_1_{etaMed;Conv;jet}
RooGaussian
Signalcsthstr_2_{etaMed;Conv;jet}
RooAddPdf
Signal_pT_{etaMed;Conv;jet}
RooBifurGauss
SignalpT_1_{etaMed;Conv;jet}
RooGaussian
SignalpT_2_{etaMed;Conv;jet}
RooGaussian
SignalpT_3_{etaMed;Conv;jet}
RooRealVar
n_Background_{etaMed;Conv;jet}
RooFormulaVar
nCat_Signal_{etaMed;Conv;jet}_reparametrized_hgg
RooRealVar
f_Signal_{etaMed;Conv;jet}
RooAddPdf
sumPdf_{etaBad;Conv;jet}_reparametrized_hgg
RooProdPdf
Background_{etaBad;Conv;jet}
Hfitter::MggBkgPdf
Background_mgg_{etaBad;Conv;jet}
RooAddPdf
Background_cosThStar_{etaBad;Conv;jet}
RooGaussian
Background_bump_cosThStar_1_{etaBad;Conv;jet}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaBad;Conv;jet}
RooAddPdf
Background_pT_{etaBad;Conv;jet}
Hfitter::HftPolyExp
Background_pT_2_{etaBad;Conv;jet}
Hfitter::HftPolyExp
Background_pT_1_{etaBad;Conv;jet}
RooProdPdf
Signal_{etaBad;Conv;jet}
RooAddPdf
Signal_mgg_{etaBad;Conv;jet}
RooCBShape
Signal_mgg_Peak_{etaBad;Conv;jet}
RooFormulaVar
mHiggsFormula_{etaBad;Conv;jet}
RooAddPdf
Signal_cosThStar_{etaBad;Conv;jet}
RooGaussian
Signalcsthstr_1_{etaBad;Conv;jet}
RooGaussian
Signalcsthstr_2_{etaBad;Conv;jet}
RooAddPdf
Signal_pT_{etaBad;Conv;jet}
RooBifurGauss
SignalpT_1_{etaBad;Conv;jet}
RooGaussian
SignalpT_2_{etaBad;Conv;jet}
RooGaussian
SignalpT_3_{etaBad;Conv;jet}
RooRealVar
n_Background_{etaBad;Conv;jet}
RooFormulaVar
nCat_Signal_{etaBad;Conv;jet}_reparametrized_hgg
RooRealVar
f_Signal_{etaBad;Conv;jet}
RooAddPdf
sumPdf_{etaGood;noConv;vbf}_reparametrized_hgg
RooProdPdf
Background_{etaGood;noConv;vbf}
Hfitter::MggBkgPdf
Background_mgg_{etaGood;noConv;vbf}
RooRealVar
xi_vbf
RooAddPdf
Background_cosThStar_{etaGood;noConv;vbf}
RooGaussian
Background_bump_cosThStar_1_{etaGood;noConv;vbf}
RooRealVar
bkg_csth01_{etaGood;vbf}
RooRealVar
bkg_csthSigma1_{etaGood;vbf}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaGood;noConv;vbf}
RooRealVar
bkg_csthPower_{etaGood;vbf}
RooRealVar
bkg_csthCoef1_{etaGood;vbf}
RooRealVar
bkg_csthCoef2_{etaGood;vbf}
RooRealVar
bkg_csthCoef4_{etaGood;vbf}
RooRealVar
bkg_csthCoef6_{etaGood;vbf}
RooRealVar
bkg_csthRelNorm1_{etaGood;vbf}
RooAddPdf
Background_pT_{etaGood;noConv;vbf}
Hfitter::HftPolyExp
Background_pT_2_{etaGood;noConv;vbf}
RooRealVar
bkg_ptPow2_vbf
RooRealVar
bkg_ptExp2_vbf
Hfitter::HftPolyExp
Background_pT_1_{etaGood;noConv;vbf}
RooRealVar
bkg_ptPow1_vbf
RooRealVar
bkg_ptExp1_vbf
RooRealVar
bkg_ptTail1Norm_vbf
RooProdPdf
Signal_{etaGood;noConv;vbf}
RooAddPdf
Signal_mgg_{etaGood;noConv;vbf}
RooCBShape
Signal_mgg_Peak_{etaGood;noConv;vbf}
RooFormulaVar
mHiggsFormula_{etaGood;noConv;vbf}
RooAddPdf
Signal_cosThStar_{etaGood;noConv;vbf}
RooGaussian
Signalcsthstr_1_{etaGood;noConv;vbf}
RooRealVar
sig_csthstr0_1_{etaGood;vbf}
RooGaussian
Signalcsthstr_2_{etaGood;noConv;vbf}
RooRealVar
sig_csthstr0_2_{etaGood;vbf}
RooRealVar
sig_csthstrRelNorm_1_{etaGood;vbf}
RooAddPdf
Signal_pT_{etaGood;noConv;vbf}
RooBifurGauss
SignalpT_1_{etaGood;noConv;vbf}
RooRealVar
sig_pt0_1_vbf
RooRealVar
sig_ptSigmaL_1_vbf
RooRealVar
sig_ptSigmaR_1_vbf
RooGaussian
SignalpT_2_{etaGood;noConv;vbf}
RooRealVar
sig_pt0_2_vbf
RooRealVar
sig_ptSigma_2_vbf
RooGaussian
SignalpT_3_{etaGood;noConv;vbf}
RooRealVar
sig_pt0_3_vbf
RooRealVar
sig_ptSigma_3_vbf
RooRealVar
sig_ptRelNorm_1_vbf
RooRealVar
sig_ptRelNorm_2_vbf
RooRealVar
n_Background_{etaGood;noConv;vbf}
RooFormulaVar
nCat_Signal_{etaGood;noConv;vbf}_reparametrized_hgg
RooRealVar
f_Signal_{etaGood;noConv;vbf}
RooAddPdf
sumPdf_{etaMed;noConv;vbf}_reparametrized_hgg
RooProdPdf
Background_{etaMed;noConv;vbf}
Hfitter::MggBkgPdf
Background_mgg_{etaMed;noConv;vbf}
RooAddPdf
Background_cosThStar_{etaMed;noConv;vbf}
RooGaussian
Background_bump_cosThStar_1_{etaMed;noConv;vbf}
RooRealVar
bkg_csth01_{etaMed;vbf}
RooRealVar
bkg_csthSigma1_{etaMed;vbf}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaMed;noConv;vbf}
RooRealVar
bkg_csthPower_{etaMed;vbf}
RooRealVar
bkg_csthCoef1_{etaMed;vbf}
RooRealVar
bkg_csthCoef2_{etaMed;vbf}
RooRealVar
bkg_csthCoef4_{etaMed;vbf}
RooRealVar
bkg_csthCoef6_{etaMed;vbf}
RooRealVar
bkg_csthRelNorm1_{etaMed;vbf}
RooAddPdf
Background_pT_{etaMed;noConv;vbf}
Hfitter::HftPolyExp
Background_pT_2_{etaMed;noConv;vbf}
Hfitter::HftPolyExp
Background_pT_1_{etaMed;noConv;vbf}
RooProdPdf
Signal_{etaMed;noConv;vbf}
RooAddPdf
Signal_mgg_{etaMed;noConv;vbf}
RooCBShape
Signal_mgg_Peak_{etaMed;noConv;vbf}
RooFormulaVar
mHiggsFormula_{etaMed;noConv;vbf}
RooAddPdf
Signal_cosThStar_{etaMed;noConv;vbf}
RooGaussian
Signalcsthstr_1_{etaMed;noConv;vbf}
RooRealVar
sig_csthstr0_1_{etaMed;vbf}
RooGaussian
Signalcsthstr_2_{etaMed;noConv;vbf}
RooRealVar
sig_csthstr0_2_{etaMed;vbf}
RooRealVar
sig_csthstrRelNorm_1_{etaMed;vbf}
RooAddPdf
Signal_pT_{etaMed;noConv;vbf}
RooBifurGauss
SignalpT_1_{etaMed;noConv;vbf}
RooGaussian
SignalpT_2_{etaMed;noConv;vbf}
RooGaussian
SignalpT_3_{etaMed;noConv;vbf}
RooRealVar
n_Background_{etaMed;noConv;vbf}
RooFormulaVar
nCat_Signal_{etaMed;noConv;vbf}_reparametrized_hgg
RooRealVar
f_Signal_{etaMed;noConv;vbf}
RooAddPdf
sumPdf_{etaBad;noConv;vbf}_reparametrized_hgg
RooProdPdf
Background_{etaBad;noConv;vbf}
Hfitter::MggBkgPdf
Background_mgg_{etaBad;noConv;vbf}
RooAddPdf
Background_cosThStar_{etaBad;noConv;vbf}
RooGaussian
Background_bump_cosThStar_1_{etaBad;noConv;vbf}
RooRealVar
bkg_csth01_{etaBad;vbf}
RooRealVar
bkg_csthSigma1_{etaBad;vbf}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaBad;noConv;vbf}
RooRealVar
bkg_csthPower_{etaBad;vbf}
RooRealVar
bkg_csthCoef1_{etaBad;vbf}
RooRealVar
bkg_csthCoef2_{etaBad;vbf}
RooRealVar
bkg_csthCoef4_{etaBad;vbf}
RooRealVar
bkg_csthCoef6_{etaBad;vbf}
RooRealVar
bkg_csthRelNorm1_{etaBad;vbf}
RooAddPdf
Background_pT_{etaBad;noConv;vbf}
Hfitter::HftPolyExp
Background_pT_2_{etaBad;noConv;vbf}
Hfitter::HftPolyExp
Background_pT_1_{etaBad;noConv;vbf}
RooProdPdf
Signal_{etaBad;noConv;vbf}
RooAddPdf
Signal_mgg_{etaBad;noConv;vbf}
RooCBShape
Signal_mgg_Peak_{etaBad;noConv;vbf}
RooFormulaVar
mHiggsFormula_{etaBad;noConv;vbf}
RooAddPdf
Signal_cosThStar_{etaBad;noConv;vbf}
RooGaussian
Signalcsthstr_1_{etaBad;noConv;vbf}
RooRealVar
sig_csthstr0_1_{etaBad;vbf}
RooGaussian
Signalcsthstr_2_{etaBad;noConv;vbf}
RooRealVar
sig_csthstr0_2_{etaBad;vbf}
RooRealVar
sig_csthstrRelNorm_1_{etaBad;vbf}
RooAddPdf
Signal_pT_{etaBad;noConv;vbf}
RooBifurGauss
SignalpT_1_{etaBad;noConv;vbf}
RooGaussian
SignalpT_2_{etaBad;noConv;vbf}
RooGaussian
SignalpT_3_{etaBad;noConv;vbf}
RooRealVar
n_Background_{etaBad;noConv;vbf}
RooFormulaVar
nCat_Signal_{etaBad;noConv;vbf}_reparametrized_hgg
RooRealVar
f_Signal_{etaBad;noConv;vbf}
RooAddPdf
sumPdf_{etaGood;Conv;vbf}_reparametrized_hgg
RooProdPdf
Background_{etaGood;Conv;vbf}
Hfitter::MggBkgPdf
Background_mgg_{etaGood;Conv;vbf}
RooAddPdf
Background_cosThStar_{etaGood;Conv;vbf}
RooGaussian
Background_bump_cosThStar_1_{etaGood;Conv;vbf}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaGood;Conv;vbf}
RooAddPdf
Background_pT_{etaGood;Conv;vbf}
Hfitter::HftPolyExp
Background_pT_2_{etaGood;Conv;vbf}
Hfitter::HftPolyExp
Background_pT_1_{etaGood;Conv;vbf}
RooProdPdf
Signal_{etaGood;Conv;vbf}
RooAddPdf
Signal_mgg_{etaGood;Conv;vbf}
RooCBShape
Signal_mgg_Peak_{etaGood;Conv;vbf}
RooFormulaVar
mHiggsFormula_{etaGood;Conv;vbf}
RooAddPdf
Signal_cosThStar_{etaGood;Conv;vbf}
RooGaussian
Signalcsthstr_1_{etaGood;Conv;vbf}
RooGaussian
Signalcsthstr_2_{etaGood;Conv;vbf}
RooAddPdf
Signal_pT_{etaGood;Conv;vbf}
RooBifurGauss
SignalpT_1_{etaGood;Conv;vbf}
RooGaussian
SignalpT_2_{etaGood;Conv;vbf}
RooGaussian
SignalpT_3_{etaGood;Conv;vbf}
RooRealVar
n_Background_{etaGood;Conv;vbf}
RooFormulaVar
nCat_Signal_{etaGood;Conv;vbf}_reparametrized_hgg
RooRealVar
f_Signal_{etaGood;Conv;vbf}
RooAddPdf
sumPdf_{etaMed;Conv;vbf}_reparametrized_hgg
RooProdPdf
Background_{etaMed;Conv;vbf}
Hfitter::MggBkgPdf
Background_mgg_{etaMed;Conv;vbf}
RooAddPdf
Background_cosThStar_{etaMed;Conv;vbf}
RooGaussian
Background_bump_cosThStar_1_{etaMed;Conv;vbf}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaMed;Conv;vbf}
RooAddPdf
Background_pT_{etaMed;Conv;vbf}
Hfitter::HftPolyExp
Background_pT_2_{etaMed;Conv;vbf}
Hfitter::HftPolyExp
Background_pT_1_{etaMed;Conv;vbf}
RooProdPdf
Signal_{etaMed;Conv;vbf}
RooAddPdf
Signal_mgg_{etaMed;Conv;vbf}
RooCBShape
Signal_mgg_Peak_{etaMed;Conv;vbf}
RooFormulaVar
mHiggsFormula_{etaMed;Conv;vbf}
RooAddPdf
Signal_cosThStar_{etaMed;Conv;vbf}
RooGaussian
Signalcsthstr_1_{etaMed;Conv;vbf}
RooGaussian
Signalcsthstr_2_{etaMed;Conv;vbf}
RooAddPdf
Signal_pT_{etaMed;Conv;vbf}
RooBifurGauss
SignalpT_1_{etaMed;Conv;vbf}
RooGaussian
SignalpT_2_{etaMed;Conv;vbf}
RooGaussian
SignalpT_3_{etaMed;Conv;vbf}
RooRealVar
n_Background_{etaMed;Conv;vbf}
RooFormulaVar
nCat_Signal_{etaMed;Conv;vbf}_reparametrized_hgg
RooRealVar
f_Signal_{etaMed;Conv;vbf}
RooAddPdf
sumPdf_{etaBad;Conv;vbf}_reparametrized_hgg
RooProdPdf
Background_{etaBad;Conv;vbf}
Hfitter::MggBkgPdf
Background_mgg_{etaBad;Conv;vbf}
RooAddPdf
Background_cosThStar_{etaBad;Conv;vbf}
RooGaussian
Background_bump_cosThStar_1_{etaBad;Conv;vbf}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaBad;Conv;vbf}
RooAddPdf
Background_pT_{etaBad;Conv;vbf}
Hfitter::HftPolyExp
Background_pT_2_{etaBad;Conv;vbf}
Hfitter::HftPolyExp
Background_pT_1_{etaBad;Conv;vbf}
RooProdPdf
Signal_{etaBad;Conv;vbf}
RooAddPdf
Signal_mgg_{etaBad;Conv;vbf}
RooCBShape
Signal_mgg_Peak_{etaBad;Conv;vbf}
RooFormulaVar
mHiggsFormula_{etaBad;Conv;vbf}
RooAddPdf
Signal_cosThStar_{etaBad;Conv;vbf}
RooGaussian
Signalcsthstr_1_{etaBad;Conv;vbf}
RooGaussian
Signalcsthstr_2_{etaBad;Conv;vbf}
RooAddPdf
Signal_pT_{etaBad;Conv;vbf}
RooBifurGauss
SignalpT_1_{etaBad;Conv;vbf}
RooGaussian
SignalpT_2_{etaBad;Conv;vbf}
RooGaussian
SignalpT_3_{etaBad;Conv;vbf}
RooRealVar
n_Background_{etaBad;Conv;vbf}
RooFormulaVar
nCat_Signal_{etaBad;Conv;vbf}_reparametrized_hgg
RooRealVar
f_Signal_{etaBad;Conv;vbf}
RooRealVar
bkg_csthPower_{etaGood;jet}
RooRealVar
bkg_csthCoef1_{etaGood;jet}
RooRealVar
bkg_csthCoef2_{etaGood;jet}
RooRealVar
bkg_csthCoef4_{etaGood;jet}
RooRealVar
bkg_csthCoef6_{etaGood;jet}
RooRealVar
bkg_ptPow2_jet
RooRealVar
bkg_ptExp2_jet
RooRealVar
bkg_ptTail1Norm_jet
RooRealVar
sig_csthstr0_1_{etaGood;jet}
RooRealVar
sig_csthstr0_2_{etaGood;jet}
RooRealVar
sig_csthstrRelNorm_1_{etaGood;jet}
RooAddPdf
sumPdf_{etaGood;Conv;jet}_reparametrized_hgg
RooProdPdf
Background_{etaGood;Conv;jet}
Hfitter::MggBkgPdf
Background_mgg_{etaGood;Conv;jet}
RooAddPdf
Background_cosThStar_{etaGood;Conv;jet}
RooGaussian
Background_bump_cosThStar_1_{etaGood;Conv;jet}
Hfitter::HftPeggedPoly
Background_poly_cosThStar_{etaGood;Conv;jet}
RooAddPdf
Background_pT_{etaGood;Conv;jet}
Hfitter::HftPolyExp
Background_pT_2_{etaGood;Conv;jet}
Hfitter::HftPolyExp
Background_pT_1_{etaGood;Conv;jet}
RooProdPdf
Signal_{etaGood;Conv;jet}
RooAddPdf
Signal_mgg_{etaGood;Conv;jet}
RooCBShape
Signal_mgg_Peak_{etaGood;Conv;jet}
RooFormulaVar
mHiggsFormula_{etaGood;Conv;jet}
RooAddPdf
Signal_cosThStar_{etaGood;Conv;jet}
RooGaussian
Signalcsthstr_1_{etaGood;Conv;jet}
RooGaussian
Signalcsthstr_2_{etaGood;Conv;jet}
RooAddPdf
Signal_pT_{etaGood;Conv;jet}
RooBifurGauss
SignalpT_1_{etaGood;Conv;jet}
RooGaussian
SignalpT_2_{etaGood;Conv;jet}
RooGaussian
SignalpT_3_{etaGood;Conv;jet}
RooRealVar
n_Background_{etaGood;Conv;jet}
RooFormulaVar
nCat_Signal_{etaGood;Conv;jet}_reparametrized_hgg
RooRealVar
f_Signal_{etaGood;Conv;jet}
2.52.5
1!
0 0.5 1 1.5 2 2.5
2!
0
0.5
1
1.5
2
2.5
1.73 GeV
(1) [14.4%]
1.73 GeV
(1) [14.4%]
2.2
1 G
eV
(2)
[13.8
%]
2.21 GeV
(2) [13.8%] 2.26 GeV
(3) [10.2%]
2.26 GeV
(3) [10.2%]
2.5
8 G
eV
(4)
[9.4
%]
2.58 GeV
(4) [9.4%]
3.4
2 G
eV
(5)
[13.6
%]
3.42 GeV
(5) [13.6%]
2.26 GeV
(6) [22.7%]
2.26 GeV
(6) [22.7%]
3.33 GeV
(7) [8.2%]
3.3
3 G
eV
(7)
[8.2
%]
1.87 GeV
(8) [7.8%]
1.87 GeV
(8) [7.8%]
1!
0 0.5 1 1.5 2 2.5
2!
0
0.5
1
1.5
2
2.5
Pseudorapidity categories (at least one converted photon)
ATLAS detector and physics performance Volume IITechnical Design Report 25 May 1999
680 19 Higgs Bosons
For an integrated luminosity of 100 fb!1, a Standard Model Higgs boson in the mass range be-tween 105 GeV and 145 GeV can be observed with a significance of more than 5" by using theH# $$ channel alone. Table 19-2 also contains the estimated significances of the H# $$ channelfor an integrated luminosity of 30 fb-1, corresponding to the first three years of LHC operation.The significances at low luminosity have been evaluated by taking the resulting improvementsin mass resolution and background rejection into account. A signal in the $$ channel can only beseen in this case with a significance of % 4" over a narrow mass range between 120 and 130 GeV.
The significances quoted in Table 19-2 are slightly higher than the ones given in the TechnicalProposal. The main reason for this is the removal of the so called pT-balance cut, which was ap-plied in order to suppress bremsstrahlung background. Although without this cut the back-ground increases, there is a net gain in the significance. Another reason is the slightly improvedmass resolution which is mainly due to a more sophisticated photon energy reconstruction, sep-arating converted and non-converted photons. These gains are somewhat offset by the higherreducible background.
As an example of signal reconstruction above background, Figure 19-4 shows the expected sig-nal from a Higgs boson with mH = 120 GeV for an integrated luminosity of 100 fb-1. The H# $$
signal is clearly visible above the smooth $$ background, which is dominated by the irreduciblecontinuum of real photon pairs.
19.2.2.2 Associated production:WH, ZH and ttH
The production of the Higgs boson in association with aW or a Z boson or with a tt pair can alsobe used to search for a low-mass Higgs boson. The production cross-section for the associatedproduction is almost a factor 50 lower than for the direct production, leading to much smallersignal rates. If the associated W/Z boson or one of the top quarks is required to decay leptoni-cally, thereby leading to final states containing one isolated lepton and two isolated photons, thesignal-to-background ratio can nevertheless be substantially improved with respect to the directproduction. In addition, the vertex position can be unambiguously determined by the leptoncharged track, resulting in better mass resolution at high luminosity than for the case of directH# $$ production.
Figure 19-4 Expected H # $$ signal for mH = 120 GeV and for an integrated luminosity of 100 fb-1. The signal
is shown on top of the irreducible background (left) and after subtraction of this background (right).
10000
12500
15000
17500
20000
105 120 135
m$$
(GeV)
Eve
nts
/ 2
GeV
0
500
1000
1500
105 120 135
m$$
(GeV)
Sig
nal-
back
grou
nd, e
vent
s /
2 G
eV
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
3-channel top combinationThe graph below represents this PDF
‣ where there are several relations between the expected means in the different channels
10
RooSimultaneous
simPdf
RooProdPdf
model_ee
RooGaussian
lumiConstraint
RooConstVar
200
RooConstVar
40
RooRealVar
Lumi
RooGaussian
alpha_EleEffConstraint
RooConstVar
0
RooConstVar
1
RooRealVar
alpha_EleEff
RooGaussian
alpha_PDFConstraint
RooRealVar
alpha_PDF
RooGaussian
alpha_modelConstraint
RooRealVar
alpha_model
RooGaussian
alpha_jesConstraint
RooRealVar
alpha_jes
RooGaussian
alpha_eesConstraint
RooRealVar
alpha_ees
RooGaussian
alpha_isrfsrConstraint
RooRealVar
alpha_isrfsr
RooGaussian
alpha_XsecConstraint
RooRealVar
alpha_Xsec
RooGaussian
alpha_DY_eeConstraint
RooRealVar
alpha_DY_ee
RooGaussian
alpha_FakeRateConstraint
RooRealVar
alpha_FakeRate
RooPoisson
Pois_ee_0
RooRealVar
obsN_0
RooAdditio
n
ee_totN_0
RooProduct
tbart_ee_overallSyst_x_HistSyst_0
LinInterpVar
tbart_ee_Hist_alpha_0
RooProduct
tbart_ee_SigXsec_x_epsilon
RooRealVar
SigXsecOverSM_comb
LinInterpVar
tbart_ee_epsilon
RooProduct
SingleTop_ee_overallSyst_x_HistSyst_0
LinInterpVar
SingleTop_ee_Hist_alpha_0
LinInterpVar
SingleTop_ee_epsilon
RooProduct
Diboson_ee_overallSyst_x_HistSyst_0
LinInterpVar
Diboson_ee_Hist_alpha_0
LinInterpVar
Diboson_ee_epsilon
RooProduct
Zll_ee_overallSyst_x_HistSyst_0
LinInterpVar
Zll_ee_Hist_alpha_0
LinInterpVar
Zll_ee_epsilon
RooProduct
Ztautau_ee_overallSyst_x_HistSyst_0
LinInterpVar
Ztautau_ee_Hist_alpha_0
LinInterpVar
Ztautau_ee_epsilon
RooProduct
fake_ee_overallSyst_x_Exp_0
RooRealVar
fake_ee_expN_0
LinInterpVar
fake_ee_epsilon
RooProdPdf
model_emu
RooGaussian
alpha_MuonEffConstraint
RooRealVar
alpha_MuonEff
RooGaussian
alpha_mesConstraint
RooRealVar
alpha_mes
RooPoisson
Pois_emu_0
RooAdditio
n
emu_totN_0
RooProduct
tbart_emu_overallSyst_x_HistSyst_0
LinInterpVar
tbart_emu_Hist_alpha_0
RooProduct
SingleTop_emu_overallSyst_x_HistSyst_0
LinInterpVar
SingleTop_emu_Hist_alpha_0
LinInterpVar
SingleTop_emu_epsilon
RooProduct
Diboson_emu_overallSyst_x_HistSyst_0
LinInterpVar
Diboson_emu_Hist_alpha_0
LinInterpVar
Diboson_emu_epsilon
RooProduct
Ztautau_emu_overallSyst_x_HistSyst_0
LinInterpVar
Ztautau_emu_Hist_alpha_0
LinInterpVar
Ztautau_emu_epsilon
RooProduct
fake_emu_overallSyst_x_Exp_0
RooRealVar
fake_emu_expN_0
LinInterpVar
fake_emu_epsilon
RooProdPdf
model_mumu
RooGaussian
alpha_DY_mumuConstraint
RooRealVar
alpha_DY_mumu
RooPoisson
Pois_mumu_0
RooAdditio
n
mumu_totN_0
RooProduct
tbart_mumu_overallSyst_x_HistSyst_0
LinInterpVar
tbart_mumu_Hist_alpha_0
RooProduct
SingleTop_mumu_overallSyst_x_HistSyst_0
LinInterpVar
SingleTop_mumu_Hist_alpha_0
LinInterpVar
SingleTop_mumu_epsilon
RooProduct
Diboson_mumu_overallSyst_x_HistSyst_0
LinInterpVar
Diboson_mumu_Hist_alpha_0
LinInterpVar
Diboson_mumu_epsilon
RooProduct
Zll_mumu_overallSyst_x_HistSyst_0
LinInterpVar
Zll_mumu_Hist_alpha_0
LinInterpVar
Zll_mumu_epsilon
RooProduct
Ztautau_mumu_overallSyst_x_HistSyst_0
LinInterpVar
Ztautau_mumu_Hist_alpha_0
LinInterpVar
Ztautau_mumu_epsilon
RooProduct
fake_mumu_overallSyst_x_Exp_0
RooRealVar
fake_mumu_expN_0
LinInterpVar
fake_mumu_epsilon
RooCategory
channelCat
5.1.3 Extending the Likelihood Function to include Multiple Bins or Channels510
One may wish to extend the likelihood function in Eq. 15 to include multiple channels (e. g. ee/µµ/eµ)511
or several jet multiplicity bins. Formally, the extension looks very similar for both cases. Let us first512
consider the case of multiple bins indexed by i. The expectation for the ith bin from the kth signal or513
background contribution is514
Nexpik = L !ik"
j
#i jk#i jk($ j)
#i jk= N
expik "
j
#i jk($ j)
#i jk. (16)
Note, that we do not add the index to $ j, because we see this as a common source of systematics which515
is common for the different bins and the different signal and background contributions. The likelihood516
function is now a product over these bins517
L(!sig,L ,$ j) = "i!bins
!
Pois(Nobsi |Nexp
i,tot)"Gaus(L |L ,!L )"j
Gaus($ j = 0|$ j, %$ j= 1)
"
. (17)
The likelihood function for multiple channels is similar, with an additional product over the multiple518
channels. The only subtlety is that k now runs over the set of signal and backgrounds specific to that519
channel. Similarly, the sources of systematics might also be different for the different channels. Leaving520
the range of the indices implicit, we arrive at521
L(!sig,L ,$ j) = "l!{ee,µµ ,eµ}
#
"i!bins
!
Pois(Nobsi |Nexp
i,tot)Gaus(L |L ,!L ) "j!syst
Gaus(0|$ j,1)
"$
. (18)
5.2 Extracting Measurements from the Profile Likelihood Ratio522
Armed with the final likelihood function in Eq. 18 and the Asimov dataset, we can now derive the ex-523
pected uncertainty on the desired cross section measurement. The likelihood function can be maximized524
to determine the maximum likelihood estimate of all the parameters !sig,L , $ j. One can then consider525
the likelihood ratio526
r(!sig) =L(!sig,L , $ j)
L(!sig,L , $ j)(19)
and the profile likelihood ratio:527
& (!sig) =L(!sig,
ˆL , ˆ$ j)
L(!sig,L , $ j)(20)
34
3 observations from data13 auxiliary measurements1 parameter of interest13 nuisance parameters
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
4-channel ATLAS Higgs combination
11
RooSimultaneous
combModel
RooAddPdf
sum_hinclusive_gg_combModel
RooAddPdf
pdf_mgg_signal_gg_combModel
RooCBShape
peakPdf_gg_combModel
RooRealVar
mHiggs_gg
RooRealVar
mRes_gg
RooRealVar
tailAlpha_gg
RooRealVar
tailN_gg
RooRealVar
invmass
RooGaussian
tailPdf_gg_combModel
RooRealVar
mTail_gg
RooRealVar
sigTail_gg
RooRealVar
mggRelNorm_gg
RooExponential
bkg_hinclusive_gg_combModel
RooRealVar
c_hinclusive_gg
RooRealVar
nbkg_hinclusive_gg
RooFormulaVar
combModel_10
RooRealVar
nsig_hinclusive_gg
RooRealVar
mu
RooProdPdf
constrainedSum_{sig}_lh_combModel
RooAddPdf
sum_{sig}_lh_combModel
RooAddPdf
sigModel_lh_combModel
AsymGauss
asymGauss1_lh_combModel
RooAddition
mean_lh_combModel
RooRealVar
offset_lh
RooRealVar
mH
RooAddition
mllEdge_lh_combModel
RooProduct
mlltemp_lh_combModel
RooRealVar
mllEdgeSlope_lh
RooRealVar
mllEdgeOffset_lh
RooRealVar
sigma1_lh
RooRealVar
mllWidth_lh
RooRealVar
mTauTau
AsymGauss
asymGauss2_lh_combModel
RooProduct
sigma2_lh
RooRealVar
rSigma12_lh
AsymGauss
asymGauss3_lh_combModel
RooProduct
sigma3_lh
RooRealVar
rSigma13_lh
RooRealVar
fgauss1_lh
RooRealVar
fgauss2_lh
RooProduct
fsig_{sig}_lh_combModel
RooRealVar
rho_lh
RooRealVar
fsigExpected_lh
RooRealVar
ratioSigEff_sig_lh
RooAddPdf
sumBkg_{sig}_lh_combModel
QCDShape
ttbarPoly_lh_combModel
RooRealVar
a1_lh
RooRealVar
a2_lh
RooRealVar
a3_lh
RooAddPdf
zjjAsymGauss_lh_combModel
AsymGauss
asymGauss1_z_lh_combModel
RooAddition
mean_z_lh
RooRealVar
mZ_lh
RooRealVar
sigma1_z_lh
RooAddition
mllEdge_z_lh
RooProduct
mlltemp_z_lh
RooRealVar
mllWidth_z_lh
AsymGauss
asymGauss2_z_lh_combModel
RooProduct
sigma2_z_lh
AsymGauss
asymGauss3_z_lh_combModel
RooProduct
sigma3_z_lh
RooRealVar
fqcd_sig_lh
RooGaussian
nTrkConstraint_{sig}_lh
RooRealVar
qcdFracNtrkConstraint_lh
RooRealVar
nTrkSigma_lh
RooProdPdf
constrainedSum_{zjj}_lh_combModel
RooAddPdf
sum_{zjj}_lh_combModel
RooProduct
fsig_{zjj}_lh_combModel
RooRealVar
ratioSigEff_zjj_lh
RooAddPdf
sumBkg_{zjj}_lh_combModel
RooRealVar
fqcd_zjj_lh
RooGaussian
nTrkConstraint_{zjj}_lh
RooProdPdf
constrainedSum_{qcd}_lh_combModel
RooAddPdf
sum_{qcd}_lh_combModel
RooProduct
fsig_{qcd}_lh_combModel
RooRealVar
ratioSigEff_qcd_lh
RooAddPdf
sumBkg_{qcd}_lh_combModel
RooRealVar
fqcd_qcd_lh
RooGaussian
nTrkConstraint_{qcd}_lh
RooAddPdf
sum_{sig}_ll_combModel
RooAddPdf
sigModel_ll_combModel
AsymGauss
asymGauss1_ll_combModel
RooAddition
mean_ll_combModel
RooRealVar
offset_ll
RooAddition
mllEdge_ll_combModel
RooProduct
mlltemp_ll_combModel
RooRealVar
mllEdgeSlope_ll
RooRealVar
mllEdgeOffset_ll
RooRealVar
sigma1_ll
RooRealVar
mllWidth_ll
AsymGauss
asymGauss2_ll_combModel
RooProduct
sigma2_ll
RooRealVar
rSigma12_ll
AsymGauss
asymGauss3_ll_combModel
RooProduct
sigma3_ll
RooRealVar
rSigma13_ll
RooRealVar
fgauss1_ll
RooRealVar
fgauss2_ll
RooProduct
fsig_{sig}_ll_combModel
RooRealVar
rho_ll
RooRealVar
fsigExpected_ll
RooRealVar
ratioSigEff_sig_ll
RooAddPdf
sumBkg_{sig}_ll_combModel
QCDShape
ttbarPoly_ll_combModel
RooRealVar
a1_ll
RooRealVar
a2_ll
RooRealVar
a3_ll
RooAddPdf
zjjAsymGauss_ll_combModel
AsymGauss
asymGauss1_z_ll_combModel
RooAddition
mean_z_ll
RooRealVar
mZ_ll
RooRealVar
sigma1_z_ll
RooAddition
mllEdge_z_ll
RooProduct
mlltemp_z_ll
RooRealVar
mllWidth_z_ll
AsymGauss
asymGauss2_z_ll_combModel
RooProduct
sigma2_z_ll
AsymGauss
asymGauss3_z_ll_combModel
RooProduct
sigma3_z_ll
RooRealVar
fqcd_sig_ll
RooAddPdf
sum_{zjj}_ll_combModel
RooProduct
fsig_{zjj}_ll_combModel
RooRealVar
ratioSigEff_zjj_ll
RooAddPdf
sumBkg_{zjj}_ll_combModel
RooRealVar
fqcd_zjj_ll
RooAddPdf
sum_{qcd}_ll_combModel
RooProduct
fsig_{qcd}_ll_combModel
RooRealVar
ratioSigEff_qcd_ll
RooAddPdf
sumBkg_{qcd}_ll_combModel
RooRealVar
fqcd_qcd_ll
RooAddPdf
dataAll_zz_combModel
RooGaussian
signal_zz_combModel
RooRealVar
sig0_zz
RooRealVar
sig1_zz
RooRealVar
mZZ
RooProduct
fsig_zz_combModel
RooRealVar
fsigExpected_zz
RooRealVar
ratioSigEff_zz
RooRealVar
rho_zz
RooAddPdf
bgAll_zz_combModel
RooZZBg
zzBg_zz_combModel
RooRealVar
zz0_zz
RooRealVar
zz1_zz
RooRealVar
zz2_zz
RooRealVar
zz3_zz
RooRealVar
zz4_zz
RooRealVar
zz5_zz
RooRealVar
zz6_zz
RooRealVar
zz7_zz
RooRealVar
zz8_zz
RooRealVar
zz9_zz
RooZbbBg
zbbBg_zz_combModel
RooRealVar
zbb0_zz
RooRealVar
zbb1_zz
RooRealVar
zbb2_zz
RooRealVar
zbb3_zz
RooRealVar
zbb4_zz
RooRealVar
zbb5_zz
RooRealVar
fZZtobb_zz
RooCategory
channel
RooSuperCategory
combModel_split_index
RooCategory
type
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
12
14
16
18
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
12
14
16
18
"!!A RooPlot of "m
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
200
400
600
800
1000
1200
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
200
400
600
800
1000
1200
"!!A RooPlot of "m
(GeV)!!m100 200 300 400 500 600 700 800
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
12
14
(GeV)!!m100 200 300 400 500 600 700 800
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
12
14
"!!A RooPlot of "m
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
12
14
16
18
20
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
12
14
16
18
20
"!!A RooPlot of "m
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
200
400
600
800
1000
1200
1400
1600
(GeV)!!m40 60 80 100 120 140 160 180
Even
ts /
( 5 G
eV )
0
200
400
600
800
1000
1200
1400
1600
"!!A RooPlot of "m
(GeV)!!m100 200 300 400 500 600 700 800
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
(GeV)!!m100 200 300 400 500 600 700 800
Even
ts /
( 5 G
eV )
0
2
4
6
8
10
"!!A RooPlot of "m
""M120 125 130 135 140
Even
ts /
( 0.4
)
1400
1600
1800
2000
2200
2400
2600
2800
3000
""M120 125 130 135 140
Even
ts /
( 0.4
)
1400
1600
1800
2000
2200
2400
2600
2800
3000
"""A RooPlot of "M
mZZ100 150 200 250 300 350 400 450 500
Even
ts /
( 4.1
2 )
0
5
10
15
20
25
30
35
40
45
mZZ100 150 200 250 300 350 400 450 500
Even
ts /
( 4.1
2 )
0
5
10
15
20
25
30
35
40
45
A RooPlot of "mZZ"
signal strength in units of SM expectation0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Proj
ectio
n of
nllW
ithC
ons
0
5
10
15
20
25
30
35
40
signal strength in units of SM expectation0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Proj
ectio
n of
nllW
ithC
ons
0
5
10
15
20
25
30
35
40
A RooPlot of "signal strength in units of SM expectation"
9 observations of continuous marks1 parameter of interest27 nuisance parameters
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
9 Channel ATLAS H->WW combination
12
top level model
parameter of interestµ =
σBR
σSMBRSMRooProdPdfatlas_w
w
RooProdPdf
atlas_ww_no_lum
i_constraint
RooPoisson
sig_em_0j_atlas_w
w_no_lum
i_constraint
RooAddition
sig_em_0j_2_atlas_w
w_no_lum
i_constraint
RooProduct
sig_em_0j_2_1_atlas_w
w_no_lum
i_constraintRooC
onstVar2.179
RooR
ealVareps_e_0j
RooR
ealVareps_m
_0j
RooR
ealVareps_j_0j
RooProduct
muTim
esLratioRooR
ealVarmu
RooR
ealVarlum
iRatio
RooProduct
sig_em_0j_2_2
RooC
onstVar4.20224
RooR
ealVarnu_w
w_0j
RooR
ealVareps_w
w_0j
RooProduct
sig_em_0j_2_3
RooC
onstVar0.29302
RooR
ealVarnu_tt_0j
RooR
ealVareps_tt_0j
RooProduct
sig_em_0j_2_4
RooC
onstVar0.308836
RooR
ealVarnu_w
j_0j
RooR
ealVareps_w
j_0j
RooR
ealVarobs_s_em
_0j
RooPoisson
sig_ee_0j_atlas_ww_no_lum
i_constraint
RooAddition
sig_ee_0j_2_atlas_ww_no_lum
i_constraint
RooProduct
sig_ee_0j_2_1_atlas_ww_no_lum
i_constraint
RooProduct
sig_ee_0j_2_1_1_atlas_ww_no_lum
i_constraintRooC
onstVar0.710392
RooProduct
sig_ee_0j_2_2
RooC
onstVar1.33425
RooProduct
sig_ee_0j_2_3
RooC
onstVar0.067641
RooProduct
sig_ee_0j_2_4
RooC
onstVar0
RooR
ealVarobs_s_ee_0j
RooPoisson
sig_mm_0j_atlas_w
w_no_lum
i_constraintRooAddition
sig_mm_0j_2_atlas_w
w_no_lum
i_constraint
RooProduct
sig_mm_0j_2_1_atlas_w
w_no_lum
i_constraint
RooProduct
sig_mm_0j_2_1_1_atlas_w
w_no_lum
i_constraint
RooC
onstVar1.42377
RooProduct
sig_mm_0j_2_2
RooC
onstVar3.05588
RooProduct
sig_mm_0j_2_3
RooC
onstVar0.141381
RooR
ealVarobs_s_m
m_0j
RooPoisson
sig_em_1j_atlas_w
w_no_lum
i_constraintRooAddition
sig_em_1j_2_atlas_w
w_no_lum
i_constraint
RooProduct
sig_em_1j_2_1_atlas_w
w_no_lum
i_constraintRooC
onstVar1.13683
RooProduct
sig_em_1j_2_2
RooC
onstVar1.52274
RooR
ealVarnu_w
w_1j
RooR
ealVareps_w
w_1j
RooProduct
sig_em_1j_2_3
RooC
onstVar1.29094
RooR
ealVarnu_tt_1j
RooR
ealVareps_tt_1j
RooProduct
sig_em_1j_2_4
RooC
onstVar0.11876
RooR
ealVarnu_w
j_1j
RooR
ealVareps_w
j_1j
RooR
ealVarobs_s_em
_1j
RooPoisson
sig_ee_1j_atlas_ww_no_lum
i_constraintRooAddition
sig_ee_1j_2_atlas_ww_no_lum
i_constraint
RooProduct
sig_ee_1j_2_1_atlas_ww_no_lum
i_constraint
RooProduct
sig_ee_1j_2_1_1_atlas_ww_no_lum
i_constraint
RooC
onstVar0.38957
RooProduct
sig_ee_1j_2_2
RooC
onstVar0.449689
RooProduct
sig_ee_1j_2_3
RooC
onstVar0.441953
RooProduct
sig_ee_1j_2_4
RooC
onstVar0.11816
RooR
ealVarobs_s_ee_1j
RooPoisson
sig_mm_1j_atlas_w
w_no_lum
i_constraint
RooAddition
sig_mm_1j_2_atlas_w
w_no_lum
i_constraint
RooProduct
sig_mm_1j_2_1_atlas_w
w_no_lum
i_constraint
RooProduct
sig_mm_1j_2_1_1_atlas_w
w_no_lum
i_constraintRooC
onstVar0.825669
RooProduct
sig_mm_1j_2_2
RooC
onstVar1.08756
RooProduct
sig_mm_1j_2_3
RooC
onstVar0.941802
RooR
ealVarobs_s_m
m_1j
RooPoisson
sig_em_2j_atlas_w
w_no_lum
i_constraint
RooAddition
sig_em_2j_2_atlas_w
w_no_lum
i_constraint
RooProduct
sig_em_2j_2_1_atlas_w
w_no_lum
i_constraint
RooC
onstVar0.219195
RooProduct
sig_em_2j_2_2
RooC
onstVar0.031047
RooR
ealVarnu_w
w_2j
RooR
ealVareps_w
w_2j
RooProduct
sig_em_2j_2_3
RooC
onstVar0.049997
RooR
ealVarnu_tt_2j
RooR
ealVareps_tt_2j
RooR
ealVarobs_s_em
_2j
RooPoisson
sig_mm_2j_atlas_w
w_no_lum
i_constraint
RooAddition
sig_mm_2j_2_atlas_w
w_no_lum
i_constraint
RooProduct
sig_mm_2j_2_1_atlas_w
w_no_lum
i_constraint
RooProduct
sig_mm_2j_2_1_1_atlas_w
w_no_lum
i_constraint
RooC
onstVar0.119561
RooProduct
sig_mm_2j_2_2
RooC
onstVar0.015524
RooProduct
sig_mm_2j_2_3
RooC
onstVar0.011287
RooR
ealVarobs_s_m
m_2j
RooPoisson
sig_ee_2j_atlas_ww_no_lum
i_constraint
RooAddition
sig_ee_2j_2_atlas_ww_no_lum
i_constraint
RooProduct
sig_ee_2j_2_1_atlas_ww_no_lum
i_constraint
RooProduct
sig_ee_2j_2_1_1_atlas_ww_no_lum
i_constraint
RooC
onstVar0.078711
RooProduct
sig_ee_2j_2_2
RooC
onstVar1e-06
RooProduct
sig_ee_2j_2_3
RooC
onstVar0.020965
RooR
ealVarobs_s_ee_2j
RooPoisson
ctr_ww_em
_0j
RooR
ealVarobs_w
w_em
_0j
RooAddition
ctr_ww_em
_0j_2
RooProduct
ctr_ww_em
_0j_2_1
RooC
onstVar11.5157
RooProduct
ctr_ww_em
_0j_2_2
RooC
onstVar1.42629
RooR
ealVareps_ttc_0j
RooProduct
ctr_ww_em
_0j_2_3
RooC
onstVar0.282082
RooR
ealVareps_w
jc_0j
RooPoisson
ctr_tt_em_0j
RooR
ealVarobs_tt_em
_0j
RooProduct
ctr_tt_em_0j_2
RooC
onstVar277.218
RooPoisson
ctr_wj_em
_0j
RooR
ealVarobs_w
j_em_0j
RooProduct
ctr_wj_em
_0j_2
RooC
onstVar43.0134
RooPoisson
ctr_ww_ee_0j
RooR
ealVarobs_w
w_ee_0j
RooAddition
ctr_ww_ee_0j_2
RooProduct
ctr_ww_ee_0j_2_1
RooC
onstVar2.43621
RooProduct
ctr_ww_ee_0j_2_2
RooC
onstVar0.443272
RooProduct
ctr_ww_ee_0j_2_3
RooPoisson
ctr_wj_ee_0j
RooR
ealVarobs_w
j_ee_0j
RooProduct
ctr_wj_ee_0j_2
RooC
onstVar12.3545
RooPoisson
ctr_ww_m
m_0j
RooR
ealVarobs_w
w_m
m_0j
RooAddition
ctr_ww_m
m_0j_2
RooProduct
ctr_ww_m
m_0j_2_1
RooC
onstVar4.79357
RooProduct
ctr_ww_m
m_0j_2_2
RooC
onstVar0.675027
RooG
aussiangaus_w
w_0j
RooC
onstVar0.073
RooR
ealVarepsM
ean_ww_0j
RooG
aussiangaus_tt_0j
RooC
onstVar1.08
RooR
ealVarepsM
ean_tt_0j
RooG
aussiangaus_w
j_0j
RooC
onstVar1
RooR
ealVarepsM
ean_wj_0j
RooG
aussiangaus_ttc_0j
RooC
onstVar0.74
RooR
ealVarepsM
ean_ttc_0j
RooG
aussiangaus_w
jc_0jRooR
ealVarepsM
ean_wjc_0j
RooG
aussiangaus_e_0j
RooC
onstVar0.01
RooR
ealVarepsM
ean_e_0j
RooG
aussiangaus_m
_0j
RooR
ealVarepsM
ean_m_0j
RooG
aussiangaus_j_0j
RooC
onstVar0.075
RooR
ealVarepsM
ean_j_0j
RooPoisson
ctr_ww_em
_1jRooR
ealVarobs_w
w_em
_1j
RooAddition
ctr_ww_em
_1j_2
RooProduct
ctr_ww_em
_1j_2_1
RooC
onstVar5.47894
RooProduct
ctr_ww_em
_1j_2_2
RooC
onstVar5.9564
RooR
ealVareps_ttc_1j
RooProduct
ctr_ww_em
_1j_2_3
RooC
onstVar0.473834
RooR
ealVareps_w
jc_1j
RooPoisson
ctr_tt_em_1j
RooR
ealVarobs_tt_em
_1j
RooProduct
ctr_tt_em_1j_2
RooC
onstVar3.89057
RooPoisson
ctr_wj_em
_1jRooR
ealVarobs_w
j_em_1j
RooProduct
ctr_wj_em
_1j_2
RooC
onstVar27.3956
RooPoisson
ctr_ww_ee_1j
RooR
ealVarobs_w
w_ee_1j
RooAddition
ctr_ww_ee_1j_2
RooProduct
ctr_ww_ee_1j_2_1
RooC
onstVar1.07473
RooProduct
ctr_ww_ee_1j_2_2
RooC
onstVar1.36674
RooProduct
ctr_ww_ee_1j_2_3
RooPoisson
ctr_wj_ee_1j
RooR
ealVarobs_w
j_ee_1j
RooProduct
ctr_wj_ee_1j_2
RooC
onstVar6.893
RooPoisson
ctr_ww_m
m_1j
RooR
ealVarobs_w
w_m
m_1j
RooAddition
ctr_ww_m
m_1j_2
RooProduct
ctr_ww_m
m_1j_2_1
RooC
onstVar2.20135
RooProduct
ctr_ww_m
m_1j_2_2
RooC
onstVar2.74799
RooG
aussiangaus_w
w_1j
RooC
onstVar0.172
RooR
ealVarepsM
ean_ww_1j
RooG
aussiangaus_tt_1j
RooC
onstVar0.52
RooR
ealVarepsM
ean_tt_1jRooG
aussiangaus_w
j_1j
RooC
onstVar0.91
RooR
ealVarepsM
ean_wj_1j
RooG
aussiangaus_ttc_1j
RooC
onstVar0.2
RooR
ealVarepsM
ean_ttc_1j
RooG
aussiangaus_w
jc_1j
RooC
onstVar0.78
RooR
ealVarepsM
ean_wjc_1j
RooPoisson
ctr_ww_em
_2j
RooR
ealVarobs_w
w_em
_2j
RooAddition
ctr_ww_em
_2j_2
RooProduct
ctr_ww_em
_2j_2_1RooC
onstVar0.463318
RooProduct
ctr_ww_em
_2j_2_2
RooC
onstVar2.6206
RooR
ealVareps_ttc_2j
RooPoisson
ctr_tt_em_2j
RooR
ealVarobs_tt_em
_2j
RooProduct
ctr_tt_em_2j_2
RooC
onstVar10.4009
RooPoisson
ctr_ww_m
m_2j
RooR
ealVarobs_w
w_m
m_2j
RooAddition
ctr_ww_m
m_2j_2
RooProduct
ctr_ww_m
m_2j_2_1
RooC
onstVar0.443022
RooProduct
ctr_ww_m
m_2j_2_2
RooC
onstVar1.02566
RooPoisson
ctr_ww_ee_2j
RooR
ealVarobs_w
w_ee_2j
RooAddition
ctr_ww_ee_2j_2
RooProduct
ctr_ww_ee_2j_2_1
RooC
onstVar0.106277
RooProduct
ctr_ww_ee_2j_2_2
RooC
onstVar0.909548
RooG
aussiangaus_w
w_2j
RooC
onstVar0.54
RooR
ealVarepsM
ean_ww_2j
RooG
aussiangaus_tt_2j
RooC
onstVar0.43
RooR
ealVarepsM
ean_tt_2j
RooG
aussiangaus_ttc_2j
RooC
onstVar0.18
RooR
ealVarepsM
ean_ttc_2j
RooG
aussianlum
iConstraint
RooR
ealVarnom
inalLumi
RooR
ealVarlum
iUncert
25 measurements from data 1 parameter of interest and 24 nuisance parameters
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Visualization of the ATLAS+CMS Workspace
13
top level model
parameter of interestµ =
σBR
σSMBRSM
RooProdPdfcom
bModel
RooProdPdf
atlas_ww_0j_atlas_com
bModel
RooProdPdf
nlc_atlas_combM
odel
RooPoisson
sig_mm_0j_nlc_atlas_com
bModel
RooAddition
sig_mm_0j_2_nlc_atlas_com
bModel
RooProduct
sig_mm_0j_2_1_nlc_atlas_com
bModel
RooProduct
sig_mm_0j_2_1_1_nlc_atlas_com
bModel
RooProduct
muTim
esLratio_atlas_combM
odel
RooR
ealVarlum
iRatio_atlas
RooR
ealVarmu
RooR
ealVareps_m
_0j_atlas
RooR
ealVareps_j_0j_atlas
RooR
ealVarmc_sig_m
m_sig_atlas
RooProduct
sig_mm_0j_2_2_atlas
RooR
ealVarnu_w
w_0j_atlas
RooR
ealVareps_w
w_0j_atlas
RooR
ealVarmc_sig_m
m_w
w_atlas
RooProduct
sig_mm_0j_2_3_atlas
RooR
ealVarnu_tt_0j_atlas
RooR
ealVareps_tt_0j_atlas
RooR
ealVarmc_sig_m
m_tt_atlas
RooR
ealVarobs_s_m
m_0j_atlas
RooPoisson
sig_em_0j_nlc_atlas_com
bModel
RooAddition
sig_em_0j_2_nlc_atlas_com
bModel
RooProduct
sig_em_0j_2_1_nlc_atlas_com
bModel
RooR
ealVareps_e_0j_atlas
RooR
ealVarmc_sig_em
_sig_atlas
RooProduct
sig_em_0j_2_2_atlas
RooR
ealVarmc_sig_em
_ww_atlas
RooProduct
sig_em_0j_2_3_atlas
RooR
ealVarmc_sig_em
_tt_atlas
RooProduct
sig_em_0j_2_4_atlas
RooR
ealVarnu_w
j_0j_atlas
RooR
ealVareps_w
j_0j_atlas
RooR
ealVarmc_sig_em
_wj_atlas
RooR
ealVarobs_s_em
_0j_atlas
RooPoisson
sig_ee_0j_nlc_atlas_combM
odelRooAddition
sig_ee_0j_2_nlc_atlas_combM
odel
RooProduct
sig_ee_0j_2_1_nlc_atlas_combM
odelRooProduct
sig_ee_0j_2_1_1_nlc_atlas_combM
odelRooR
ealVarmc_sig_ee_sig_atlas
RooProduct
sig_ee_0j_2_2_atlasRooR
ealVarmc_sig_ee_w
w_atlas
RooProduct
sig_ee_0j_2_3_atlasRooR
ealVarmc_sig_ee_tt_atlas
RooProduct
sig_ee_0j_2_4_atlas
RooC
onstVar0
RooR
ealVarobs_s_ee_0j_atlas
RooPoisson
ctr_ww_m
m_0j_atlas
RooR
ealVarobs_w
w_m
m_0j_atlas
RooAddition
ctr_ww_m
m_0j_2_atlas
RooProduct
ctr_ww_m
m_0j_2_1_atlas
RooR
ealVarmc_w
w_ctrl_m
m_w
w_atlas
RooProduct
ctr_ww_m
m_0j_2_2_atlas
RooR
ealVareps_ttc_0j_atlas
RooR
ealVarmc_w
w_ctrl_m
m_tt_atlas
RooPoisson
ctr_wj_em
_0j_atlas
RooR
ealVarobs_w
j_em_0j_atlas
RooProduct
ctr_wj_em
_0j_2_atlas
RooR
ealVarmc_w
j_ctrl_em_w
j_atlas
RooPoisson
ctr_tt_em_0j_atlas
RooR
ealVarobs_tt_em
_0j_atlas
RooProduct
ctr_tt_em_0j_2_atlas
RooR
ealVarmc_tt_ctrl_tt_atlas
RooPoisson
ctr_ww_em
_0j_atlas
RooR
ealVarobs_w
w_em
_0j_atlas
RooAddition
ctr_ww_em
_0j_2_atlas
RooProduct
ctr_ww_em
_0j_2_1_atlasRooR
ealVarmc_w
w_ctrl_em
_ww_atlas
RooProduct
ctr_ww_em
_0j_2_2_atlasRooR
ealVarmc_w
w_ctrl_em
_tt_atlas
RooProduct
ctr_ww_em
_0j_2_3_atlas
RooR
ealVareps_w
jc_0j_atlas
RooR
ealVarmc_w
w_ctrl_em
_wj_atlas
RooPoisson
ctr_wj_ee_0j_atlas
RooR
ealVarobs_w
j_ee_0j_atlas
RooProduct
ctr_wj_ee_0j_2_atlas
RooR
ealVarmc_w
j_ctrl_ee_wj_atlas
RooPoisson
ctr_ww_ee_0j_atlas
RooR
ealVarobs_w
w_ee_0j_atlas
RooAddition
ctr_ww_ee_0j_2_atlas
RooProduct
ctr_ww_ee_0j_2_1_atlas
RooR
ealVarmc_w
w_ctrl_ee_w
w_atlas
RooProduct
ctr_ww_ee_0j_2_2_atlas
RooR
ealVarmc_w
w_ctrl_ee_tt_atlas
RooProduct
ctr_ww_ee_0j_2_3_atlas
RooG
aussiangaus_w
w_0j_atlas
RooC
onstVar0.073
RooR
ealVarepsM
ean_ww_0j_atlas
RooG
aussiangaus_w
jc_0j_atlas
RooR
ealVarepsM
ean_wjc_0j_atlas
RooC
onstVar1
RooG
aussiangaus_tt_0j_atlas
RooC
onstVar1.08
RooR
ealVarepsM
ean_tt_0j_atlas
RooG
aussiangaus_ttc_0j_atlas
RooC
onstVar0.74
RooR
ealVarepsM
ean_ttc_0j_atlas
RooG
aussiangaus_w
j_0j_atlasRooR
ealVarepsM
ean_wj_0j_atlas
RooG
aussiangaus_e_0j_atlas
RooC
onstVar0.01
RooR
ealVarepsM
ean_e_0j_atlas
RooG
aussiangaus_m
_0j_atlas
RooR
ealVarepsM
ean_m_0j_atlas
RooG
aussiangaus_j_0j_atlas
RooC
onstVar0.075
RooR
ealVarepsM
ean_j_0j_atlas
RooG
aussianlum
iConstraint_atlas
RooR
ealVarnom
inalLumi_atlas
RooR
ealVarlum
iUncert_atlas
RooProdPdf
modelN
uis_cms_com
bModel
RooProdPdf
model_cm
s_combM
odel
RooPoisson
model_bin1_cm
s_combM
odelRooForm
ulaVaryield_tot_bin1_cm
s_combM
odelRooForm
ulaVaryield_bin1_cat0_cm
s
RooR
ealVarnexp_bin1_cat0_cm
s
RooR
ealVarerr16_bin1_cat0_cm
s
RooR
ealVarx16_cm
s
RooR
ealVarerr25_bin1_cat0_cm
s
RooR
ealVarx25_cm
s
RooR
ealVarerr27_bin1_cat0_cm
s
RooR
ealVarx27_cm
s
RooR
ealVarerr30_bin1_cat0_cm
s
RooR
ealVarx30_cm
s
RooR
ealVarerr31_bin1_cat0_cm
s
RooR
ealVarx31_cm
s
RooR
ealVarerr33_bin1_cat0_cm
s
RooR
ealVarx33_cm
s
RooR
ealVarerr35_bin1_cat0_cm
s
RooR
ealVarx35_cm
s
RooR
ealVarerr36_bin1_cat0_cm
s
RooR
ealVarx36_cm
s
RooR
ealVarerr37_bin1_cat0_cm
s
RooR
ealVarx37_cm
s
RooForm
ulaVaryield_bkg_bin1_cm
s
RooForm
ulaVaryield_bin1_cat1_cm
sRooR
ealVarnexp_bin1_cat1_cm
s
RooR
ealVarerr1_bin1_cat1_cm
s
RooR
ealVarx1_cm
s
RooR
ealVarerr3_bin1_cat1_cm
s
RooR
ealVarx3_cm
s
RooR
ealVarerr26_bin1_cat1_cm
s
RooR
ealVarx26_cm
s
RooForm
ulaVaryield_bin1_cat2_cm
s
RooR
ealVarnexp_bin1_cat2_cm
s
RooR
ealVarerr5_bin1_cat2_cm
s
RooR
ealVarx5_cm
s
RooR
ealVarerr17_bin1_cat2_cm
s
RooR
ealVarx17_cm
s
RooR
ealVarerr26_bin1_cat2_cm
s
RooR
ealVarerr28_bin1_cat2_cm
s
RooR
ealVarx28_cm
s
RooR
ealVarerr35_bin1_cat2_cm
s
RooForm
ulaVaryield_bin1_cat3_cm
s
RooR
ealVarnexp_bin1_cat3_cm
s
RooR
ealVarerr7_bin1_cat3_cm
s
RooR
ealVarx7_cm
s
RooR
ealVarerr8_bin1_cat3_cm
s
RooR
ealVarx8_cm
s
RooR
ealVarerr9_bin1_cat3_cm
s
RooR
ealVarx9_cm
s
RooR
ealVarerr18_bin1_cat3_cm
s
RooR
ealVarx18_cm
s
RooR
ealVarerr25_bin1_cat3_cm
s
RooR
ealVarerr35_bin1_cat3_cm
s
RooForm
ulaVaryield_bin1_cat4_cm
s
RooR
ealVarnexp_bin1_cat4_cm
s
RooR
ealVarerr10_bin1_cat4_cm
s
RooR
ealVarx10_cm
s
RooR
ealVarerr13_bin1_cat4_cm
s
RooR
ealVarx13_cm
s
RooR
ealVarerr26_bin1_cat4_cm
s
RooR
ealVarerr35_bin1_cat4_cm
s
RooR
ealVarerr36_bin1_cat4_cm
sRooForm
ulaVaryield_bin1_cat5_cm
s
RooR
ealVarnexp_bin1_cat5_cm
s
RooR
ealVarerr26_bin1_cat5_cm
s
RooR
ealVarerr29_bin1_cat5_cm
s
RooR
ealVarx29_cm
s
RooR
ealVarerr31_bin1_cat5_cm
s
RooR
ealVarerr33_bin1_cat5_cm
s
RooR
ealVarerr35_bin1_cat5_cm
s
RooR
ealVarerr36_bin1_cat5_cm
s
RooR
ealVarerr37_bin1_cat5_cm
s
RooForm
ulaVaryield_bin1_cat6_cm
s
RooR
ealVarnexp_bin1_cat6_cm
s
RooR
ealVarerr26_bin1_cat6_cm
s
RooR
ealVarerr29_bin1_cat6_cm
s
RooR
ealVarerr31_bin1_cat6_cm
s
RooR
ealVarerr33_bin1_cat6_cm
s
RooR
ealVarerr35_bin1_cat6_cm
s
RooR
ealVarerr36_bin1_cat6_cm
s
RooR
ealVarerr37_bin1_cat6_cm
s
RooR
ealVarnobs_bin1_cm
s
RooPoisson
model_bin2_cm
s_combM
odel
RooForm
ulaVaryield_tot_bin2_cm
s_combM
odel
RooForm
ulaVaryield_bin2_cat0_cm
s
RooR
ealVarnexp_bin2_cat0_cm
s
RooR
ealVarerr19_bin2_cat0_cm
s
RooR
ealVarx19_cm
s
RooR
ealVarerr25_bin2_cat0_cm
s
RooR
ealVarerr27_bin2_cat0_cm
s
RooR
ealVarerr30_bin2_cat0_cm
s
RooR
ealVarerr32_bin2_cat0_cm
s
RooR
ealVarx32_cm
s
RooR
ealVarerr34_bin2_cat0_cm
s
RooR
ealVarx34_cm
s
RooR
ealVarerr35_bin2_cat0_cm
s
RooR
ealVarerr36_bin2_cat0_cm
s
RooR
ealVarerr37_bin2_cat0_cm
s
RooForm
ulaVaryield_bkg_bin2_cm
s
RooForm
ulaVaryield_bin2_cat1_cm
s
RooR
ealVarnexp_bin2_cat1_cm
s
RooR
ealVarerr2_bin2_cat1_cm
s
RooR
ealVarx2_cm
s
RooR
ealVarerr4_bin2_cat1_cm
s
RooR
ealVarx4_cm
s
RooR
ealVarerr26_bin2_cat1_cm
s
RooForm
ulaVaryield_bin2_cat2_cm
s
RooR
ealVarnexp_bin2_cat2_cm
s
RooR
ealVarerr6_bin2_cat2_cm
s
RooR
ealVarx6_cm
s
RooR
ealVarerr20_bin2_cat2_cm
s
RooR
ealVarx20_cm
s
RooR
ealVarerr26_bin2_cat2_cm
s
RooR
ealVarerr28_bin2_cat2_cm
s
RooR
ealVarerr35_bin2_cat2_cm
s
RooForm
ulaVaryield_bin2_cat3_cm
s
RooR
ealVarnexp_bin2_cat3_cm
s
RooR
ealVarerr7_bin2_cat3_cm
s
RooR
ealVarerr8_bin2_cat3_cm
s
RooR
ealVarerr9_bin2_cat3_cm
s
RooR
ealVarerr21_bin2_cat3_cm
s
RooR
ealVarx21_cm
s
RooR
ealVarerr25_bin2_cat3_cm
s
RooR
ealVarerr35_bin2_cat3_cm
s
RooForm
ulaVaryield_bin2_cat4_cm
sRooR
ealVarnexp_bin2_cat4_cm
s
RooR
ealVarerr11_bin2_cat4_cm
s
RooR
ealVarx11_cm
s
RooR
ealVarerr14_bin2_cat4_cm
s
RooR
ealVarx14_cm
s
RooR
ealVarerr26_bin2_cat4_cm
s
RooR
ealVarerr35_bin2_cat4_cm
s
RooR
ealVarerr36_bin2_cat4_cm
s
RooForm
ulaVaryield_bin2_cat5_cm
sRooR
ealVarnexp_bin2_cat5_cm
s
RooR
ealVarerr26_bin2_cat5_cm
s
RooR
ealVarerr29_bin2_cat5_cm
s
RooR
ealVarerr32_bin2_cat5_cm
s
RooR
ealVarerr34_bin2_cat5_cm
s
RooR
ealVarerr35_bin2_cat5_cm
s
RooR
ealVarerr36_bin2_cat5_cm
s
RooR
ealVarerr37_bin2_cat5_cm
s
RooForm
ulaVaryield_bin2_cat6_cm
s
RooR
ealVarnexp_bin2_cat6_cm
s
RooR
ealVarerr26_bin2_cat6_cm
s
RooR
ealVarerr29_bin2_cat6_cm
s
RooR
ealVarerr32_bin2_cat6_cm
s
RooR
ealVarerr34_bin2_cat6_cm
s
RooR
ealVarerr35_bin2_cat6_cm
s
RooR
ealVarerr36_bin2_cat6_cm
s
RooR
ealVarerr37_bin2_cat6_cm
s
RooR
ealVarnobs_bin2_cm
s
RooPoisson
model_bin3_cm
s_combM
odelRooForm
ulaVaryield_tot_bin3_cm
s_combM
odel
RooForm
ulaVaryield_bin3_cat0_cm
s
RooR
ealVarnexp_bin3_cat0_cm
s
RooR
ealVarerr22_bin3_cat0_cm
s
RooR
ealVarx22_cm
s
RooR
ealVarerr25_bin3_cat0_cm
s
RooR
ealVarerr27_bin3_cat0_cm
s
RooR
ealVarerr30_bin3_cat0_cm
s
RooR
ealVarerr31_bin3_cat0_cm
s
RooR
ealVarerr32_bin3_cat0_cm
s
RooR
ealVarerr33_bin3_cat0_cm
s
RooR
ealVarerr34_bin3_cat0_cm
s
RooR
ealVarerr35_bin3_cat0_cm
s
RooR
ealVarerr36_bin3_cat0_cm
s
RooR
ealVarerr37_bin3_cat0_cm
s
RooForm
ulaVaryield_bkg_bin3_cm
s
RooForm
ulaVaryield_bin3_cat1_cm
s
RooR
ealVarnexp_bin3_cat1_cm
s
RooR
ealVarerr1_bin3_cat1_cm
s
RooR
ealVarerr4_bin3_cat1_cm
s
RooR
ealVarerr26_bin3_cat1_cm
s
RooForm
ulaVaryield_bin3_cat2_cm
s
RooR
ealVarnexp_bin3_cat2_cm
s
RooR
ealVarerr5_bin3_cat2_cm
s
RooR
ealVarerr23_bin3_cat2_cm
s
RooR
ealVarx23_cm
s
RooR
ealVarerr26_bin3_cat2_cm
s
RooR
ealVarerr28_bin3_cat2_cm
s
RooR
ealVarerr35_bin3_cat2_cm
s
RooForm
ulaVaryield_bin3_cat3_cm
s
RooR
ealVarnexp_bin3_cat3_cm
s
RooR
ealVarerr7_bin3_cat3_cm
s
RooR
ealVarerr8_bin3_cat3_cm
s
RooR
ealVarerr9_bin3_cat3_cm
s
RooR
ealVarerr24_bin3_cat3_cm
s
RooR
ealVarx24_cm
s
RooR
ealVarerr25_bin3_cat3_cm
s
RooR
ealVarerr35_bin3_cat3_cm
s
RooForm
ulaVaryield_bin3_cat4_cm
s
RooR
ealVarnexp_bin3_cat4_cm
s
RooR
ealVarerr12_bin3_cat4_cm
s
RooR
ealVarx12_cm
s
RooR
ealVarerr15_bin3_cat4_cm
s
RooR
ealVarx15_cm
s
RooR
ealVarerr26_bin3_cat4_cm
s
RooR
ealVarerr35_bin3_cat4_cm
s
RooR
ealVarerr36_bin3_cat4_cm
s
RooForm
ulaVaryield_bin3_cat5_cm
s
RooR
ealVarnexp_bin3_cat5_cm
s
RooR
ealVarerr26_bin3_cat5_cm
s
RooR
ealVarerr29_bin3_cat5_cm
s
RooR
ealVarerr31_bin3_cat5_cm
s
RooR
ealVarerr32_bin3_cat5_cm
s
RooR
ealVarerr33_bin3_cat5_cm
s
RooR
ealVarerr34_bin3_cat5_cm
s
RooR
ealVarerr35_bin3_cat5_cm
s
RooR
ealVarerr36_bin3_cat5_cm
s
RooR
ealVarerr37_bin3_cat5_cm
s
RooForm
ulaVaryield_bin3_cat6_cm
s
RooR
ealVarnexp_bin3_cat6_cm
s
RooR
ealVarerr26_bin3_cat6_cm
s
RooR
ealVarerr29_bin3_cat6_cm
s
RooR
ealVarerr31_bin3_cat6_cm
s
RooR
ealVarerr32_bin3_cat6_cm
s
RooR
ealVarerr33_bin3_cat6_cm
s
RooR
ealVarerr34_bin3_cat6_cm
s
RooR
ealVarerr35_bin3_cat6_cm
s
RooR
ealVarerr36_bin3_cat6_cm
s
RooR
ealVarerr37_bin3_cat6_cm
s
RooR
ealVarnobs_bin3_cm
s
RooProdPdf
nuisPrior_cms
RooG
aussianpdf_x1_cm
s
RooG
aussianpdf_x2_cm
s
RooG
aussianpdf_x3_cm
s
RooG
aussianpdf_x4_cm
s
RooG
aussianpdf_x5_cm
s
RooG
aussianpdf_x6_cm
s
RooG
aussianpdf_x7_cm
s
RooG
aussianpdf_x8_cm
s
RooG
aussianpdf_x9_cm
s
RooG
aussianpdf_x10_cm
s
RooG
aussianpdf_x11_cm
s
RooG
aussianpdf_x12_cm
s
RooG
aussianpdf_x13_cm
s
RooG
aussianpdf_x14_cm
s
RooG
aussianpdf_x15_cm
s
RooG
aussianpdf_x16_cm
s
RooG
aussianpdf_x17_cm
s
RooG
aussianpdf_x18_cm
s
RooG
aussianpdf_x19_cm
s
RooG
aussianpdf_x20_cm
s
RooG
aussianpdf_x21_cm
s
RooG
aussianpdf_x22_cm
s
RooG
aussianpdf_x23_cm
s
RooG
aussianpdf_x24_cm
s
RooG
aussianpdf_x25_cm
s
RooG
aussianpdf_x26_cm
s
RooG
aussianpdf_x27_cm
s
RooG
aussianpdf_x28_cm
s
RooG
aussianpdf_x29_cm
s
RooG
aussianpdf_x30_cm
s
RooG
aussianpdf_x31_cm
s
RooG
aussianpdf_x32_cm
s
RooG
aussianpdf_x33_cm
s
RooG
aussianpdf_x34_cm
s
RooG
aussianpdf_x35_cm
s
RooG
aussianpdf_x36_cm
s
RooG
aussianpdf_x37_cm
s
ATLAS part
CMS part
The full model has 12 observables and~50 parameters
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Graphical Models
14
Bayesian vs. Frequentist
Graphical models
Gaussianity and isotropy
Lyons’ wishlist
Bayesian information
3 6
1 5 7
2 4
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� �✲
❅❅❘
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❅❅❘
✲❅
❅❘❅
❅❘
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Directed Markov means
f (x) = f (x1)f (x2 | x1)f (x3 | x1)f (x4 | x2)
× f (x5 | x2, x3)f (x6 | x3, x5)f (x7 | x4, x5, x6).
Theory exists for deriving all conditional independencies and
exploiting local structure in graph for gross computational
simplifications in complex models. Has been successfully exploited
in AI, machine learning, and Bayesian statistics.
Steffen Lauritzen University of Oxford Statistical respondent
P
Pon
non s b
Poff
noff !
non
s b
noff !
non
s b
noff
!
P(data | parameters) P(parameters|data)
s b
!
?
P(parameters)
Given all these graphs, it’s not surprising that one might think there’s an application for Graphical Models‣ graphs are different, but let’s
discuss connection
Graph of on/off model
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Raw Measurements to Interpretation
From the raw LHC data, the experiments estiamate (“measure”) several interesting quantities (like masses of particles)
‣ ideally the likelihood function for those quantities is provided
These quantities are not the parameters of the fundamental theory, but the are usually functions of the fundamental parameters
‣ an entire industry has emerged that interprets these observations in terms of a specific theory (see Roberto Trotta’s talk for an example)
15
LHC DataData Modeling
Experimentalist provide p.d.f.s that
relate masses, cross-sections, etc. to
observables including systematics are
modeled using RooFit/RooStats
Likelihood
Function
Digital publishing via
RooFit/RooStats
Workspace saved
in portable .root file
Interpretation
Tools like ZFitter & SFitter interface to
likelihood function to extract
fundamental Lagrangian parameters
Fundamental
Lagrangian
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Our theories are parametrized in some form convenient for our underlying quantum field theories. But this parametrization is somewhat arbitrary, and
‣ phenomenology nearly constant in large regions and changes quickly in others. ‣ It would be useful to efficiently sample this space efficiently‣ eg... uniform in fisher information metric
Some applications of the Fisher information metric
16
Weak Boson Fusion H ! !!
jj
H
!! h
l
MissingET is the dominant experimental issue
Unexpected complications from finely seg-mented calorimeter and noise suppression
Several GeV of bias in MissingET if one simplycuts all cells with E < 2"noise
Translates into bias on m!!
Complementarity of h ! !! and H ! !! al-lows this channel to cover most of the MSSMHiggs plane.
5
10
15
20
25
30
80 100 120 140 160MA[GeV]
tan!
LEP2: e+e"!Zh
LHC(40fb-1):VV!H!## VV!h!##
Plehn, et. al hep-ph/9911385
August 23, 2005
ALCWS, Snowmass, 2005
Higgs at the LHC & SLHC (page 6) Kyle Cranmer
Brookhaven National Lab
Information Geometry
Amari considered the Fisher Information Matrix gij as a metric on a
Manifold M parametrized by !:
gij(!) =
!
dxf!(x)
"
" log f!(x)
"!i
# "
" log f!(x)
"!j
#
Example:Consider Gaussians G(x; µ,!) as a 2-d Manifoldparametrized by " = (µ,!)
the geometry is isotropic and negatively curved
Natural Learning Rules correspond to geodesics
on the Manifold M .
Can lead to exponentially faster rates of convergence!
April 11, 2005
EFI High Energy Physics Seminar
Modern Data Analysis Techniques
for High Energy Physics (page 46)
Kyle Cranmer
Brookhaven National Laboratory
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Spinoffs from the Asimov idea
17
gij(θ) = E��
∂
∂θilnL(θ)
� �∂
∂θjlnL(θ)
����� θ
�.
gij(θ) ≈�
∂
∂θilnLA(θ)
� �∂
∂θjlnLA(θ)
�
Reconstructed Mass
Num
ber
of
Even
ts
Reconstructed Mass
Num
ber
of
Even
ts
Calculating the Fisher info. matrix requires an expectation over possible data.
This also provides a convenient algorithm determining for Jeffreys’s prior numerically, but I know their are issues with numerics and improper priors.
In many problems, this is too computationally expensive to be useful.
We found that the curvature of the likelihood function on the Asimov data gives a very good estimate of gij
Last night, Earl L. and Richard L. helped us see that this curvature of this single Asimov dataset can be seen as a numerical integration for calculating the expectation of the curvature.
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
The Asimov datasetThe name of the “Asimov” data set is inspired by the short story Franchise, by Isaac Asimov.
“Multivac picked you as the most representative this year. Not the smartest, or the strongest, or the luckiest, but just the most representative. Now we don’t question Multivac, do we?”
Coincidentally, the story takes place in 2008, when we started to formalize the properties of our “Asimov” Dataset
18
Reconstructed Mass
Num
ber
of
Even
ts
Glen Cowan, KC, Eilam Gross, Ofer Vitellshttp://arxiv.org/abs/1007.1727
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Implicit vs. Explicit systematicsIn some cases, effect of systematics is explicitly parametrized with nuisance parameters.
19
x
f(x|α,β)
nominal
β variationα variation
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Implicit vs. Explicit systematicsIn other cases, one simply has a flexible model parametrized by υ, which is flexible enough to incorporate the systematic effects‣ so dependence on α,β (previously identified with specific
systematic effects) is implicit
20
x
f(x|!)
nominal
! variation
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
x
f(x|!)
nominal
! variation
x
f(x|α,β)
nominal
β variationα variation
Implicit vs. Explicit systematicsA problem arises when one wants to combine these two measurements knowing that the systematic effects of α,β are correlated between the two measurements‣ but there is no explicit handle on α,β in the implicit model
‣ in some cases this may reduce to reparametrization‣ in some cases effect of several systematic effects may produce a degenerate
deformation of the shape, so it’s not clear the dimensionality of the parametrization is even the same
Basic idea is to add a term that summarizes the correlation between the parameters...
‣ what is the procedure for determining this term, especially if we want to maintain a frequentist interpretation
21
P (α,β, ν)
ν(α,β)
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Conditioning & Look elsewhereLook-elsewhere effect:‣ location of peak is meaningless in null model‣ several possibilities for background to fluctuate
● typical approach is to understand and/or correct for “trials” factor (Bonferroni, talks by Eilam and Ofer2, ...)
Is there an alternative approach based on conditioning:‣ eg., what is p-value for a peak this large in the background for
the ensemble where the biggest peak is located at this point.
22
Reconstructed Mass
Num
ber
of
Even
ts
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
Extras
23
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
RooFit: A data modeling toolkit
24Wouter Verkerke, UCSB
Building realistic models
– Composition (‘plug & play’)
– Convolution
g(x;m,s)m(y;a0,a1)
=
! =
g(x,y;a0,a1,s)Possible in any PDF
No explicit support in PDF code needed
Wouter Verkerke, UCSB
Building realistic models
• Complex PDFs be can be trivially composed using operator classes
– Addition
– Multiplication
+ =
* =
Wouter Verkerke, UCSB
Parameters of composite PDF objects
RooAddPdf
sum
RooGaussian
gauss1RooGaussian
gauss2RooArgusBG
argusRooRealVar
g1fracRooRealVar
g2frac
RooRealVar
xRooRealVar
sigmaRooRealVar
mean1
RooRealVar
mean2RooRealVar
argparRooRealVar
cutoff
RooArgSet *paramList = sum.getParameters(data) ;
paramList->Print("v") ;
RooArgSet::parameters:
1) RooRealVar::argpar : -1.00000 C
2) RooRealVar::cutoff : 9.0000 C
3) RooRealVar::g1frac : 0.50000 C
4) RooRealVar::g2frac : 0.10000 C
5) RooRealVar::mean1 : 2.0000 C
6) RooRealVar::mean2 : 3.0000 C
7) RooRealVar::sigma : 1.0000 C
The parameters of sumare the combined parametersof its components
A major tool at BaBar. Fit complicated models with >100 parameters!
Kyle Cranmer (NYU)
Center for Cosmology and Particle Physics
BIRS, July 13, 2010
m(ll) [GeV]
0 20 40 60 80 100 120 140 160 180 200
-1E
ntr
ies/4
GeV
/ 1 f
b
-10
0
10
20
30
40
50
/ ndf 2 40.11 / 45
Prob 0.679
Endpoint 1.399± 99.66
Norm. 0.02563± -0.3882
Smearing 1.339± 2.273
/ ndf 2 40.11 / 45
Prob 0.679
Endpoint 1.399± 99.66
Norm. 0.02563± -0.3882
Smearing 1.339± 2.273
ATLAS
Supersymmetric Mass MeasurementsHere is a graphical representation of a measurement used for supersymmetric parameter estimation
‣ Functions‣ Parameters of Interest‣ Nuisance Parameters‣ Observable
25
P
TriangleSmearing /
Transfer
P (mll|mχ01, mχ0
2, ml, σ) = Triangle(mtrue
ll , medgell (mχ0
1, mχ0
2, m))⊕ Smearing(mtrue
ll , mll)
medgell = mχ0
2
����1−�
ml
mχ02
�2�
1−�
mχ01
ml
�2
Helmholtz Ian Hinchliffe 12/1/07 28
!"#"$%&'()*+'*%,#-.)$"*'
! /%**%#,'"#"$,+0'*.-12"'3$'4)-*%'*.-12"'5(6
! 7%,8'9".'3$'2":.3#';)2.%:2%&%.+0'7"-<+'319"&.'="&-+%#,'%#'="."&.3$
! >%#";-.%&'*.$)&.)$"'?$3;'&-*&-="'="&-+*0'3#2+'%?'*:"&.$);'%*'@&3;:-&.A0'B""=*'23#,'&8-%#*'.3'C3$D'C"220';-#+'EF'%#<32<"=
! /3$"';3="2'=":"#="#.'G"H-;:2"*I
" 53#,'2%<"='319"&.*
! (2":.3#*'3$',-),%#3*'%#'!/(E
! !2)%#3*'%#'*:2%.'(J(K
!""
!
#$ "$ !$!02
$ !01$
% %
!""
!
#$ "$ !$!02
$ !01$
% %
Kyle Cranmer (NYU) Cosmostats, July 28, 2009
Center for Cosmology and Particle PhysicsMatrix Element Method as a Parametrization
26
February 25, 2009 Daniel Whiteson/UC Irvine Slide 7
Matrix-element likelihood: Calculate probability directly
P(event z | SM) = P (z | process A) + P(z | process B) + ….
where
P( z | A ) = ! dy |MA|2 fpfp fTF(y,z)
Parton(y) to detector(z) transfer function (TF) describes parton-shower and detector response in parametrized form (Issue 2)
Matrix-element*PDFs for process A (Issue 2)
Integration over parton-level quantities
= d"#/dz
Prediction via Monte Carlo Simulation
The enormous detectors are still being constructed, but we have detailedsimulations of the detectors response.
L(x|H0) =W
W
Hµ+
µ!
!
The advancements in theoretical predictions, detector simulation, tracking,calorimetry, triggering, and computing set the bar high for equivalentadvances in our statistical treatment of the data.
September 13, 2005
PhyStat2005, OxfordStatistical Challenges of the LHC (page 6) Kyle Cranmer
Brookhaven National Laboratory
Diagrams by MadGraph g g -> u u~ e- e+ ap gp
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 1
1
2
3
4
5
6
7
8
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 2
1
2
3
4
5
6
7
8
u
u
e
e
ap
gp gp
qp
zp
lp
graph 3
1
2
3
4
5
6
7
8
u
u
e
e ap
gp
gp
qp
zp
lp
gp
graph 4
1
2
3
4
5
6
7
8
u
u
e
e ap
gp
gp
qp
zp
lp
gp
graph 5
1
2
3
4
5
6
7
8
u
u
e
e ap
gp gp
qp
zp
lp
graph 6
1
2
3
4
5
6
7
8
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 7
1
2 3
4
5
6
7
8
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 8
1
2
3
4
5
6
7
8
m(ll) [GeV]
0 20 40 60 80 100 120 140 160 180 200
-1E
ntr
ies/4
GeV
/ 1 f
b
-10
0
10
20
30
40
50
/ ndf 2 40.11 / 45
Prob 0.679
Endpoint 1.399± 99.66
Norm. 0.02563± -0.3882
Smearing 1.339± 2.273
/ ndf 2 40.11 / 45
Prob 0.679
Endpoint 1.399± 99.66
Norm. 0.02563± -0.3882
Smearing 1.339± 2.273
ATLASP( | µ, ν) =
With the same di-lepton mass distribution, we can either:‣ relate edge according to:
‣ incorporate matrix element techniques
medgell = mχ0
2
����1−�
ml
mχ02
�2�
1−�
mχ01
ml
�2
Kyle Cranmer (NYU) Cosmostats, July 28, 2009
Center for Cosmology and Particle PhysicsMatrix Element Method as a Parametrization
26
February 25, 2009 Daniel Whiteson/UC Irvine Slide 7
Matrix-element likelihood: Calculate probability directly
P(event z | SM) = P (z | process A) + P(z | process B) + ….
where
P( z | A ) = ! dy |MA|2 fpfp fTF(y,z)
Parton(y) to detector(z) transfer function (TF) describes parton-shower and detector response in parametrized form (Issue 2)
Matrix-element*PDFs for process A (Issue 2)
Integration over parton-level quantities
= d"#/dz
Prediction via Monte Carlo Simulation
The enormous detectors are still being constructed, but we have detailedsimulations of the detectors response.
L(x|H0) =W
W
Hµ+
µ!
!
The advancements in theoretical predictions, detector simulation, tracking,calorimetry, triggering, and computing set the bar high for equivalentadvances in our statistical treatment of the data.
September 13, 2005
PhyStat2005, OxfordStatistical Challenges of the LHC (page 6) Kyle Cranmer
Brookhaven National Laboratory
Diagrams by MadGraph g g -> u u~ e- e+ ap gp
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 1
1
2
3
4
5
6
7
8
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 2
1
2
3
4
5
6
7
8
u
u
e
e
ap
gp gp
qp
zp
lp
graph 3
1
2
3
4
5
6
7
8
u
u
e
e ap
gp
gp
qp
zp
lp
gp
graph 4
1
2
3
4
5
6
7
8
u
u
e
e ap
gp
gp
qp
zp
lp
gp
graph 5
1
2
3
4
5
6
7
8
u
u
e
e ap
gp gp
qp
zp
lp
graph 6
1
2
3
4
5
6
7
8
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 7
1
2 3
4
5
6
7
8
u
u
e
e
ap
gp
gp qp
zp
lp gp
graph 8
1
2
3
4
5
6
7
8
m(ll) [GeV]
0 20 40 60 80 100 120 140 160 180 200
-1E
ntr
ies/4
GeV
/ 1 f
b
-10
0
10
20
30
40
50
/ ndf 2 40.11 / 45
Prob 0.679
Endpoint 1.399± 99.66
Norm. 0.02563± -0.3882
Smearing 1.339± 2.273
/ ndf 2 40.11 / 45
Prob 0.679
Endpoint 1.399± 99.66
Norm. 0.02563± -0.3882
Smearing 1.339± 2.273
ATLASP( | µ, ν) =
With the same di-lepton mass distribution, we can either:‣ relate edge according to:
‣ incorporate matrix element techniques● naturally, could include more
kinematic info -> more power.
medgell = mχ0
2
����1−�
ml
mχ02
�2�
1−�
mχ01
ml
�2