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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
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
!
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Comments on these lectures
5
Fred James gave a terrific series of lectures. Largely based on principles, focused on comparison of Bayesian & Frequentist
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Comments on these lectures
5
Fred James gave a terrific series of lectures. Largely based on principles, focused on comparison of Bayesian & Frequentist
Louis Lyons, also gave terrific lectures, basic principles plus some useful examples
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Comments on these lectures
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
Fred James gave a terrific series of lectures. Largely based on principles, focused on comparison of Bayesian & Frequentist
Louis Lyons, also gave terrific lectures, basic principles plus some useful examples
Glen Cowan gave lectures for CERN summer school (slightly lower level) & great academic training lectures on multivariate algorithms
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Comments on these lectures
5
Fred James gave a terrific series of lectures. Largely based on principles, focused on comparison of Bayesian & Frequentist
Louis Lyons, also gave terrific lectures, basic principles plus some useful examples
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Comments on these lectures
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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.
# $# P(B) is the prior or marginal probability of B, and acts as a normalizing constant.
13
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
... in pictures (from Bob Cousins)
14
P, Conditional P, and Derivation of Bayes’ Theorem
in Pictures
A B
Whole space
P(A) = P(B) =
P(A B) =
P(B|A) = P(A|B) =
P(B) ! P(A|B) = ! =
P(A " B) =
P(A) ! P(B|A) = ! = = P(A " B)
= P(A " B)
! P(B|A) = P(A|B) ! P(B) / P(A) Bob Cousins, CMS, 2008 7
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
... in pictures (from Bob Cousins)
14
P, Conditional P, and Derivation of Bayes’ Theorem
in Pictures
A B
Whole space
P(A) = P(B) =
P(A B) =
P(B|A) = P(A|B) =
P(B) ! P(A|B) = ! =
P(A " B) =
P(A) ! P(B|A) = ! = = P(A " B)
= P(A " B)
! P(B|A) = P(A|B) ! P(B) / P(A) Bob Cousins, CMS, 2008 7
Don’t forget about “Whole space” . I will drop it from the notation typically, but occasionally it is important.
!
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Louis’s Example
15
16
!"#$%&%'()*+,-."""""""""!"#()*+,-'$%&%.!
()*+,-""/"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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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:
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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) ?
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
Bob’s example of Bayes’ theorem
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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) ?
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)
Note, this use of Bayes’ theorem is fine for Frequentist
17
Bob’s example of Bayes’ theorem
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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)
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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)
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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 )
Question: one observes 8 events, what is P(Higgs | N=8) ?
19
An different example of Bayes’ theorem
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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 )
Question: one observes 8 events, what is P(Higgs | N=8) ?
Answer: Cannot be determined from the given information!
! 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
An different example of Bayes’ theorem
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
a little on Information Theory
22
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
How much information in this message?
Information Theory
23
1000110101001011! "# $16 entries
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
How much information in this message?
What about this one
Information Theory
23
1000110101001011! "# $16 entries
01010101010101! "# $16 entries
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
How much information in this message?
What about this one
... and this one?
Information Theory
23
1000110101001011! "# $16 entries
abcdabcdabcdabcd! "# $16 entries
01010101010101! "# $16 entries
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
How much information in this message?
! 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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Cumulative Density FunctionsOften useful to use a cumulative distribution:
! 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:
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Cumulative Density FunctionsOften useful to use a cumulative distribution:
! 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:
! similar to relationship of total and di"erential cross section:
f(E) =1!
"!
"E
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Cumulative Density FunctionsOften useful to use a cumulative distribution:
! 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:
! similar to relationship of total and di"erential cross section:
f(E, !) =1"
#2"
#E#!
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Cumulative Density FunctionsOften useful to use a cumulative distribution:
! 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:
! similar to relationship of total and di"erential cross section:
f(E, !) =1"
#2"
#E#!
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Bayes’ Theorem: the continuous case
27
2008 CERN Summer Student Lectures on Statistics 14G. Cowan
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!)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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, !)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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, !)
G
x mu sigma
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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, !)
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}
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Parametric vs. Non-Parametric PDFs
Alternatively, one can consider non-parametric pdfs
29
From empirical data, one has empirical PDF
femp =1N
N!
i
!(x! xi)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Parametric vs. Non-Parametric PDFs
Alternatively, one can consider non-parametric pdfs
30
or, one can make a histogram
fw,shist(x) =
1N
!
i
hw,si
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Parametric vs. Non-Parametric PDFs
Alternatively, one can consider non-parametric pdfs
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
Parametric vs. Non-Parametric PDFs
Alternatively, one can consider non-parametric pdfs
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)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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]
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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)
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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)]
2008 CERN Summer Student Lectures on Statistics 20G. Cowan
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
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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”
Monday, February 2, 2009
Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009
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
Monday, February 2, 2009