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Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009

Statistics for Particle Physics

Kyle Cranmer

New York University

1

Monday, February 2, 2009


Statistics plays a vital role in science, it is the way that we:

! quantify our knowledge and uncertainty

! communicate results of experiments

Big questions:

! testing theories, measure or exclude parameters, etc.

! how do we make decisions

! how do we get the most out of our data

! how do we incorporate uncertainties

Statistics is a very big field, and it is not possible to cover everything in 4 hours. In these talks I will try to:

! explain some fundamental ideas & prove a few things

! enrich what you already know

! expose you to some new ideas

I will try to go slowly, because if you are not following the logic, then it is not very interesting.

! Please feel free to ask questions and interrupt at any time

2

Introduction



Further ReadingBy physicists, for physicists

G. Cowan, Statistical Data Analysis, Clarendon Press, Oxford, 1998.

R.J.Barlow, A Guide to the Use of Statistical Methods in the Physical Sciences, John Wiley, 1989;

F. James, Statistical Methods in Experimental Physics, 2nd ed., World Scientific, 2006;

! W.T. Eadie et al., North-Holland, 1971 (1st ed., hard to find);

S.Brandt, Statistical and Computational Methods in Data Analysis, Springer, New York, 1998.

L.Lyons, Statistics for Nuclear and Particle Physics, CUP, 1986.

My favorite statistics book by a statistician:

3

!


http://www.pp.rhul.ac.uk/~cowan/sda/

http://www.pp.rhul.ac.uk/~cowan/sda/


Other lecturesFred James’s lectures

Glen Cowan’s lectures

Louis Lyons

Bob Cousins gave a CMS lecture, may give it more publicly

The PhyStat conference series at PhyStat.org:

4

http://www.desy.de/~acatrain/

http://www.pp.rhul.ac.uk/~cowan/stat_cern.html

http://preprints.cern.ch/cgi-bin/setlink?base=AT&categ=Academic_Training&id=AT00000799

http://indico.cern.ch/conferenceDisplay.py?confId=a063350


http://phystat.org

http://phystat.org










Comments on these lectures

5

Fred James gave a terrific series of lectures. Largely based on principles, focused on comparison of Bayesian & Frequentist




5


Louis Lyons, also gave terrific lectures, basic principles plus some useful examples




52008 CERN Summer Student Lectures on Statistics 5G. Cowan

Dealing with uncertainty In particle physics there are various elements of uncertainty:

theory is not deterministicquantum mechanics

random measurement errorspresent even without quantum effects

things we could know in principle but don’te.g. from limitations of cost, time, ...

We can quantify the uncertainty using PROBABILITY

6 Glen Cowan Multivariate Statistical Methods in Particle Physics

Finding an optimal decision boundaryMaybe select events with “cuts”:

xi < cixj < cj

Or maybe use some other type of decision boundary:

Goal of multivariate analysis is to do this in an “optimal” way.

H0 H0

H0

H1

H1H1



Glen Cowan gave lectures for CERN summer school (slightly lower level) & great academic training lectures on multivariate algorithms




5



Glen Cowan gave lectures for CERN summer school (slightly lower level) & great academic training lectures on multivariate algorithms

Bob Cousins gave a very comprehensive lecture to CMS on “statistics in theory”

“Statistics in Theory”*: Prelude to “Statistics in Practice”

Bob Cousins, UCLABob Cousins, UCLA

CMS Statistics Tutorial Series

May 8, 2008

*Background for sound work and for avoiding unsound statements.

Bob Cousins, CMS, 2008 1




6

So what will be the theme of these lectures?

Definitely not a cook book, I want to convey the fundamental concepts in a fairly general setting.

But I don’t want to spend time on special cases or contrived examples. I want to address the challenges of the LHC.

I also don’t want to discuss purely theoretical results if they aren’t directly applicable. However, there are many theoretical results that provide an insightful bound.

In particular, I’m mainly interested in discovery and measurement, but I will touch on goodness of fit and limits.

I also hope to sprinkle the lectures with advanced topics and expose you to some modern approaches and unsolved problems.

In theory, there is no difference between theory and practice; In practice, there is.- Chuck Reid

There is nothing more practical than a good theory.- James C. Maxwell



OutlineLecture 1:

! How we use statistics

! Probability axioms, Bayes vs. Frequentist, from discrete to continuous

! Parametric and non-parametric probability density functions

! Shannon and Fisher Information, correlation, information geometry, Cramér-Rao bound

! A word on subjective and “objective” Bayesian priors

Lecture 2

! Hypothesis testing in the frequentist setting

! The Neyman-Pearson lemma (with a simple proof)

! Decision theory: utility, risk, priors, and game theory

! Contrast hypothesis testing to goodness of fit tests with some warnings

! Related comments on multivariate algorithms

! Matrix element techniques vs. the black box

7



OutlineLecture 3:

! The Neyman-Construction (illustrated)

! Inverted hypothesis tests: A dictionary for limits (intervals)

! Coverage as a calibration for our statistical device

! Compound hypotheses, nuisance parameters, & similar tests

! Systematics, Systematics, Systematics

Lecture 4:

! Generalizing our procedures to include systematics

! Eliminating nuisance parameters: profiling and marginalization

! Introduction to ancillary statistics & conditioning

! High dimensional models, Markov Chain Monte Carlo, and Hierarchical Bayes

! The look elsewhere e"ect and false discovery rate

8



Lecture 1

9



Broadly speaking, we use statistical techniques for a few main purposes:

! Point estimation: what is the best estimate of a particular parameter

# eg. measurement of the Z boson mass

! Confidence Intervals: regions representing an allowable range of a parameter (in a way to be made precise later)

# eg. 95% contours, upper-limits, lower-limits

! Hypothesis Testing: choosing between two (or more) hypotheses

# eg. Discover the Higgs, Discover SUSY, reject standard model

! Goodness-of-fit: quantify how well the data agrees with a particular model

! Data reduction: how to reduce the raw data while loosing minimal information tha is useful for our ultimate goal

In a broader context, there are related issues:

! Decision making: how do we make decisions in the face of uncertainty

! Where does the role of an experimentalist end? How does this impact how we publish our results? or how we make decisions?

10

How We Use Statistics



Axioms of Probability

These Axioms are a mathematical starting point for probability and statistics

1.probability for every element, E, is non-negative

2.probability for the entire space of possibilities is 1

3.if elements Ei are disjoint, probability is additive

Consequences:

11

Kolmogorov

axioms"(1933)



Different definitions of Probability

12

http://plato.stanford.edu/archives/sum2003/entries/probability-interpret/#3.1

|!" | #$|2 =12

Frequentist

! defined as limit of long term frequency

! probability of rolling a 3 := limit of (# rolls with 3 / # trials)

# you don’t need an infinite sample for definition to be useful

# sometimes ensemble doesn’t exist

• eg. P(Higgs mass = 120 GeV), P(it will snow tomorrow)

! Intuitive if you are familiar with Monte Carlo methods

! compatible with interpretation of probability in Quantum Mechanics (though some argue this point). Probability to measure spin projected on x-axis if spin of beam is polarized along +z

Subjective Bayesian

! Probability is a degree of belief (personal, subjective)

# can be made quantitative based on betting odds

# most people’s subjective probabilities are not coherent and do not obey laws of probability





Bayes’ TheoremBayes’ theorem relates the conditional and marginal probabilities of events A & B

# $# P(A) is the prior probability or marginal probability of A. It is "prior" in the sense

that it does not take into account any information about B.

# $# P(A|B) is the conditional probability of A, given B. It is also called the posterior

probability because it is derived from or depends upon the specified value of B.

# $# P(B|A) is the conditional probability of B given A.

# $# P(B) is the prior or marginal probability of B, and acts as a normalizing constant.

13


http://en.wikipedia.org/wiki/Normalizing_constant

http://en.wikipedia.org/wiki/Prior_probability

http://en.wikipedia.org/wiki/Prior_probability

http://en.wikipedia.org/wiki/Marginal_probability

http://en.wikipedia.org/wiki/Marginal_probability

http://en.wikipedia.org/wiki/Conditional_probability

http://en.wikipedia.org/wiki/Conditional_probability

http://en.wikipedia.org/wiki/Posterior_probability




http://en.wikipedia.org/wiki/Normalizing_constant


... in pictures (from Bob Cousins)

14

P, Conditional P, and Derivation of Bayes’ Theorem

in Pictures

A B

Whole space

P(A) = P(B) =

P(A B) =

P(B|A) = P(A|B) =

P(B) ! P(A|B) = ! =

P(A " B) =

P(A) ! P(B|A) = ! = = P(A " B)

= P(A " B)

! P(B|A) = P(A|B) ! P(B) / P(A) Bob Cousins, CMS, 2008 7



... in pictures (from Bob Cousins)

14

P, Conditional P, and Derivation of Bayes’ Theorem

in Pictures

A B

Whole space

P(A) = P(B) =

P(A B) =

P(B|A) = P(A|B) =

P(B) ! P(A|B) = ! =

P(A " B) =

P(A) ! P(B|A) = ! = = P(A " B)

= P(A " B)

! P(B|A) = P(A|B) ! P(B) / P(A) Bob Cousins, CMS, 2008 7

Don’t forget about “Whole space” . I will drop it from the notation typically, but occasionally it is important.

!



Louis’s Example

15

16

!"#$%&%'()*+,-."""""""""!"#()*+,-'$%&%.!

()*+,-""/"0%1*"+,"2*0%1*

$%&%"/"""3,*45%5&"+,"5+&"3,*45%5&

!"#3,*45%5&"'"2*0%1*."6"78

9:&

!"#2*0%1*"'"3,*45%5&.";;;78



Bob’s Example

A b-tagging algorithm gives a curve like this

One wants to decide where to cut and to optimize analysis

! For some point on the curve you have:

# P(btag| b-jet), i.e., e%ciency for tagging b’s

# P(btag| not a b-jet), i.e., e%ciency for background

16

The Factory 10

Signal efficiency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ba

ckg

rou

nd

reje

cti

on

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Signal efficiency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ba

ckg

rou

nd

reje

cti

on

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

MVA Method:

Fisher

MLP

LikelihoodD

PDERS

Cuts

RuleFit

HMatrix

BDT

Likelihood

Background rejection versus Signal efficiency

Figure 5:

Code Example 6:



Now that you know:

! P(btag| b-jet), i.e., e%ciency for tagging b’s

! P(btag| not a b-jet), i.e., e%ciency for background

Question: Given a selection of jets tagged as b-jets, what fraction of them are b-jets?

! I.e., what is P(b-jet | btag) ?

17

Bob’s example of Bayes’ theorem



Now that you know:





Answer: Cannot be determined from the given information!

! Need to know P(b-jet): fraction of all jets that are b-jets.

! Then Bayes’ Theorem inverts the conditionality:

# P(b-jet | btag) !P(btag|b-jet) P(b-jet)

17




Now that you know:






! Need to know P(b-jet): fraction of all jets that are b-jets.

! Then Bayes’ Theorem inverts the conditionality:

# P(b-jet | btag) !P(btag|b-jet) P(b-jet)

Note, this use of Bayes’ theorem is fine for Frequentist

17




Bayesian vs. Frequentist

In short, Frequentist are always restricted to statements related to

! P(Data | Theory) (deductive reasoning)

! the data is considered random

! each point in the “Theory” space is treated independently # (no notion of probability in the “Theory” space)

18








Bayesians can address questions of the form:

! P(Theory | Data) ! P(Data | Theory) P(Theory)

# intuitively what we want to know (inductive reasoning)

! but it requires a prior on the Theory# [short discussion subjective vs. empirical Bayes goes here]

18








Bayesians can address questions of the form:

! P(Theory | Data) ! P(Data | Theory) P(Theory)

# intuitively what we want to know (inductive reasoning)

! but it requires a prior on the Theory# [short discussion subjective vs. empirical Bayes goes here]

Later I will discuss the “Likelihood Principle” and Likelihood-based analysis: it’s a third approach to statistical inference

18


http://en.wikipedia.org/wiki/Statistical_inference

http://en.wikipedia.org/wiki/Statistical_inference


An analysis is developed to search for the Higgs boson

! background expectation is 0.1 events

# you know P(N | no Higgs)

! signal expectation is 10 events

# you know P(N | Higgs )

19

An different example of Bayes’ theorem








Question: one observes 8 events, what is P(Higgs | N=8) ?

19









Question: one observes 8 events, what is P(Higgs | N=8) ?


! Need in addition: P(Higgs)

# no ensemble! no frequentist notion of P(Higgs)

# prior based on degree-of-belief would work, but it is subjective. This is why some people object to Bayesian statistics for particle physics

19




A Joke

20

“Bayesians address the question everyone is interested in, by using assumptions no-one believes”

“Frequentists use impeccable logic to deal with an issue of no interest to anyone”

- P. G. Hamer



Some personal history

21

Archbishop of Canterbury Thomas

Cranmer (born: 1489, executed:

1556) author of the “Book of

Common Prayer”

Two centuries later (when this Book

had become an official prayer book of

the Church of England) Thomas Bayes

was a non-conformist minister

(Presbyterian) who refused to use

Cranmer$s book



a little on Information Theory

22



How much information in this message?

Information Theory

23

1000110101001011! "# $16 entries




What about this one

Information Theory

23

1000110101001011! "# $16 entries

01010101010101! "# $16 entries




What about this one

... and this one?

Information Theory

23

1000110101001011! "# $16 entries

abcdabcdabcdabcd! "# $16 entries

01010101010101! "# $16 entries



Information Theory

24

H(X) = !!

x!X

p(x) log p(x)

0 0.5 1.00

0.5

1.0

Pr(X = 1)

H(X

)

1

1000110101001011! "# $16 entries


! 16 bits? (bit is unit when log is base 2)

! it depends on probabilities of 0,1

In 1870’s Boltzman and Gibbs defined entropy:

In 1948, Calude Shannon uses entropy as a centerpiece of his “Mathematical Theory of Communication” eg. information theory

! information maximized when pi all equal



Probability Density Functions

When dealing with continuous random variables, need to introduce the notion of a Probability Density Function (PDF... not parton distribution function)

Note, is NOT a probability

Equivalent of second axiom...

25

P (x ! [x, x + dx]) = f(x)dx

! !

"!f(x)dx = 1

f(x)



Cumulative Density FunctionsOften useful to use a cumulative distribution:

! in 1-dimension:

26

x-3 -2 -1 0 1 2 3

F(x)

0

0.2

0.4

0.6

0.8

1

x-3 -2 -1 0 1 2 3

f(x)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

! x

!"f(x#)dx# = F (x)




! in 1-dimension:

26

f(x) =!F (x)

!x

x-3 -2 -1 0 1 2 3

F(x)

0

0.2

0.4

0.6

0.8

1

x-3 -2 -1 0 1 2 3

f(x)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

! x

!"f(x#)dx# = F (x)

! alternatively, define density as partial of cumulative:




! in 1-dimension:

26

f(x) =!F (x)

!x

x-3 -2 -1 0 1 2 3

F(x)

0

0.2

0.4

0.6

0.8

1

x-3 -2 -1 0 1 2 3

f(x)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

! x

!"f(x#)dx# = F (x)


! similar to relationship of total and di"erential cross section:

f(E) =1!

"!

"E




! in 1-dimension:

26

f(x) =!F (x)

!x

x-3 -2 -1 0 1 2 3

F(x)

0

0.2

0.4

0.6

0.8

1

x-3 -2 -1 0 1 2 3

f(x)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

! x

!"f(x#)dx# = F (x)


! similar to relationship of total and di"erential cross section:

f(E, !) =1"

#2"

#E#!



Bayes’ Theorem: the continuous case

27


Bayesian Statistics ! general philosophy In Bayesian statistics, use subjective probability for hypotheses:

posterior probability, i.e., after seeing the data

prior probability, i.e.,before seeing the data

probability of the data assuming hypothesis H (the likelihood)

normalization involves sum over all possible hypotheses

Bayes’ theorem has an “if-then” character: If your priorprobabilities were ! (H), then it says how these probabilitiesshould change in the light of the data.

No unique prescription for priors (subjective!)



Parametric vs. Non-Parametric PDFs

Many familiar pdfs are considered parametric

! eg. a Gaussian is parametrized by

! defines a family of functions

! allows one to make inference about parameters

! some examples have very complicated parametric pdfs

28

G(x|µ, !) (µ, !)









28

G(x|µ, !) (µ, !)

G

x mu sigma









28

G(x|µ, !) (µ, !)

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


RooRealVar


RooRealVar


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


RooRealVar

sig_pt0_2_noJets

RooRealVar

sig_ptSigma_2_noJets

RooGaussian


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}


Background_poly_cosThStar_{etaMed;noConv;noJets}

RooRealVar

bkg_csthPower_{etaMed;noJets}

RooRealVar

bkg_csthCoef1_{etaMed;noJets}

RooRealVar


RooRealVar


RooRealVar


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


RooGaussian


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}


Background_poly_cosThStar_{etaBad;noConv;noJets}

RooRealVar

bkg_csthPower_{etaBad;noJets}

RooRealVar

bkg_csthCoef1_{etaBad;noJets}

RooRealVar


RooRealVar


RooRealVar


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


RooGaussian


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}


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


RooGaussian


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}


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


RooGaussian


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}


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


RooGaussian


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}


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


RooRealVar

sig_pt0_2_jet

RooRealVar

sig_ptSigma_2_jet

RooGaussian


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}


Background_poly_cosThStar_{etaMed;noConv;jet}

RooRealVar

bkg_csthPower_{etaMed;jet}

RooRealVar

bkg_csthCoef1_{etaMed;jet}

RooRealVar


RooRealVar


RooRealVar


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


RooGaussian


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}


Background_poly_cosThStar_{etaBad;noConv;jet}

RooRealVar

bkg_csthPower_{etaBad;jet}

RooRealVar

bkg_csthCoef1_{etaBad;jet}

RooRealVar


RooRealVar


RooRealVar


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


RooGaussian


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}


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


RooGaussian


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}


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


RooGaussian


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}


Background_poly_cosThStar_{etaGood;noConv;vbf}

RooRealVar

bkg_csthPower_{etaGood;vbf}

RooRealVar

bkg_csthCoef1_{etaGood;vbf}

RooRealVar


RooRealVar


RooRealVar


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


RooRealVar

sig_pt0_2_vbf

RooRealVar

sig_ptSigma_2_vbf

RooGaussian


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}


Background_poly_cosThStar_{etaMed;noConv;vbf}

RooRealVar

bkg_csthPower_{etaMed;vbf}

RooRealVar

bkg_csthCoef1_{etaMed;vbf}

RooRealVar


RooRealVar


RooRealVar


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


RooGaussian


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}


Background_poly_cosThStar_{etaBad;noConv;vbf}

RooRealVar

bkg_csthPower_{etaBad;vbf}

RooRealVar

bkg_csthCoef1_{etaBad;vbf}

RooRealVar


RooRealVar


RooRealVar


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


RooGaussian


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}


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


RooGaussian


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}


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


RooGaussian


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}


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


RooGaussian


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


RooRealVar


RooRealVar


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}


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


RooGaussian


RooRealVar

n_Background_{etaGood;Conv;jet}

RooFormulaVar

nCat_Signal_{etaGood;Conv;jet}_reparametrized_hgg

RooRealVar

f_Signal_{etaGood;Conv;jet}



x-3 -2 -1 0 1 2 3

f(x)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Alternatively, one can consider non-parametric pdfs

29

From empirical data, one has empirical PDF

femp =1N

N!

i

!(x! xi)



x-3 -2 -1 0 1 2 3

f(x)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7



30

or, one can make a histogram

fw,shist(x) =

1N

!

i

hw,si





31

x-3 -2 -1 0 1 2 3

f(x)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

but they depend on bin width and starting position

fw,shist(x) =

1N

!

i

hw,si





32

x-3 -2 -1 0 1 2 3

f(x)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

“Average Shifted Histogram” minimizes e"ect of binning

fwASH(x) =

1N

N!

i

Kw(x! xi)



Kernel Estimation

“the data is the model”

Adaptive Kernel estimation puts wider kernels in regions of low probability

Used at LEP for describing pdfs from Monte Carlo (KEYS)

33

Neural Network Output

Pro

bab

ilit

y D

ensi

tyf̂1(x) =n

!

i

1

nh(xi)K

"

x ! xi

h(xi)

#

h(xi) =

"

4

3

#1/5 $

!

f̂0(xi)n!1/5

Kernel estimation is the generalization of Average Shifted Histograms

K.Cranmer, Comput.Phys.Commun. 136 (2001). [hep-ex/0011057]



Multivariate PDFs

34

Max Baak 6

Correlations

! 2-d projection of pdf from previous slide.

! RooNDKeys pdfautomatically models (fine) correlations between observables ...

ttbar sample

higgs sample

Kernel Estimation has a nice generalizations to higher dimensions

! practical limit is about 5-d due to curse of dimensionality

Max Baak has coded N-dim KEYS pdf described in Comput.Phys.Commun. 136 (2001) in RooFit.

These pdfs have been used as the basis for a multivariate discrimination technique called “PDE”

D(!x) =fs(!x)

fs(!x) + fb(!x)



Correlation / Covariance

Correlation is a common way to describe how one variable depends on another

! however, it only captures the lowest order of dependence between variables, and

35

X

Y

X

Y

X

Y

X

Y

cov[x, y] = Vxy = E[(x! µx)(y ! µy)]


Covariance and correlationDefine covariance cov[x,y] (also use matrix notation Vxy) as

Correlation coefficient (dimensionless) defined as

If x, y, independent, i.e., , then

! x and y, ‘uncorrelated’

N.B. converse not always true.Monday, February 2, 2009


Propagation of errors

The Covariance matrix plays a central rôle in propagation of errors from to

but remember, that this is only the first-order in the Taylor expansion

36

x y



Mutual Information

A more general notion of ‘correlation’ comes from Mutual Information:

! it is symmetric: I(X;Y) = I(Y;X)

! if and only if X,Y totally independent: I(X;Y)=0

! possible for X,Y to be uncorrelated, but not independent

37X

YMutual Information doesn’t seem to be used much within HEP, but it seems quite useful



Lecture 1:

! How we use statistics

! Probability axioms, Bayes vs. Frequentist, from discrete to continuous

! Parametric and non-parametric probability density functions

! Shannon and Fisher Information, correlation, information geometry, Cramér-Rao bound

! A word on subjective and “objective” Bayesian priors

38

Remaining topics for “Lecture 1”



Next TimeLecture 2

! Hypothesis testing in the frequentist setting

! The Neyman-Pearson lemma (with a simple proof)

! Decision theory: utility, risk, priors, and game theory

! Contrast hypothesis testing to goodness of fit tests with some warnings

! Related comments on multivariate algorithms

! Matrix element techniques vs. the black box

Lecture 3:

! The Neyman-Construction (illustrated)

! Inverted hypothesis tests: A dictionary for limits (intervals)

! Coverage as a calibration for our statistical device

! Compound hypotheses, nuisance parameters, & similar tests

! Systematics, Systematics, Systematics

Lecture 4:

! Generalizing our procedures to include systematics

! Eliminating nuisance parameters: profiling and marginalization

! Introduction to ancillary statistics & conditioning

! High dimensional models, Markov Chain Monte Carlo, and Hierarchical Bayes

! The look elsewhere e"ect and false discovery rate

39


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