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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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

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2, ml, σ) = Triangle(mtrue

ll , medgell (mχ0

1, mχ0

2, m))⊕ Smearing(mtrue

ll , mll)

medgell = mχ0

2

����1−�

ml

mχ02

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1−�

mχ01

ml

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Helmholtz Ian Hinchliffe 12/1/07 28

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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

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u

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gp gp

qp

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lp

graph 6

1

2

3

4

5

6

7

8

u

u

e

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ap

gp

gp qp

zp

lp gp

graph 7

1

2 3

4

5

6

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8

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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