Lecture 1 Harrison B Prosper Florida State University European School of HighEnergy Physics Parádfürdő Hungary 5 18 June 2013 1 Outline Lecture 1 Descriptive Statistics ID: 779471
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Slide1
Practical Statistics for Particle PhysicistsLecture 1
Harrison B. ProsperFlorida State UniversityEuropean School of High-Energy PhysicsParádfürdő, Hungary 5 – 18 June, 2013
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Slide2OutlineLecture 1Descriptive Statistics
Probability & Likelihood Lecture 2The Frequentist Approach The Bayesian Approach Lecture 3
Analysis Example
2
Slide3Descriptive Statistics
Slide4Descriptive Statistics – 1Definition: A statistic
is any function of the data X.Given a sample X = x1, x2, … xN, it is often of interest to compute statistics such as the sample average
and the
sample variance
In any analysis, it is good practice to study
ensemble averages
, denoted by < … >, of relevant statistics
4
Slide5Descriptive Statistics – 2Ensemble Average
MeanErrorBiasVarianceMean Square Error5
Slide66Descriptive Statistics –
3 The MSE is the most widely used measure of closeness of anensemble of statistics {x} to the true value μ The root mean square (RMS) is
Exercise 1
:
Show this
Slide7Descriptive Statistics – 4Consider the ensemble average of the
sample variance7
Slide8Descriptive Statistics – 5The ensemble average of the sample variance
has a negative bias of –V / N 8
Exercise 2
:
Show this
Slide9Descriptive Statistics – SummaryThe sample average
is an unbiased estimateof the ensemble averageThe sample variance is a biased estimateof the ensemble variance9
Slide10Probability
Slide1111Probability – 1
Basic Rules 1. P(A) ≥ 0 2. P(A) = 1 if A is true 3. P(A) = 0 if A is falseSum Rule 4. P(A+B) = P(A) + P(B) if AB is false *Product Rule 5. P(AB) = P(A|B) P(B) *
*A+B = A or B, AB = A and B, A|B = A given that B is true
Slide12Probability – 2
By definition, the conditional probability of A given B isP(A) is the probability of A withoutrestriction. P(A|B) is the probability of A when we
restric
t
to the conditions under
which
B
is true.
12
A
B
A
B
Slide13Fromwe deduceBayes’ Theorem
:13A
B
A
B
Probability –
3
Slide14A and B are mutually exclusive
if P(AB) = 0A and B are exhaustive if P(A) + P(B) = 1
Theorem
14
Probability –
4
Exercise 3
: Prove theorem
Slide15Probability Binomial & Poisson Distributions
Slide16Binomial & Poisson Distributions – 1A Bernoulli trial has two outcomes:
S = success or F = failure. Example: Each collision between protons at the LHC is a Bernoulli trial in which something interesting happens (S) or does not (F). 16
Slide17Binomial & Poisson Distributions – 2Let p be the probability of a success, which is assumed to be the
same at each trial. Since S and F are exhaustive, the probability of a failure is 1 – p. For a given order O of n trails, the probability Pr(k,O|n) of exactly k successes and n – k failures is 17
Slide18Binomial & Poisson Distributions – 3If the order O
of successes and failures is irrelevant, we can eliminate the order from the problem integrating over all possible ordersThis yields the binomial distributionwhich is sometimes written as 18
Slide19Binomial & Poisson Distributions – 3We can prove that the mean number of successes a is
a = p n. Suppose that the probability, p, of a success is very small,then, in the limit p → 0 and n → ∞, such that a is constant, Binomial(k, n, p
)
→
Poisson(k
,
a
).
The Poisson distribution is generally regarded as a good model for a counting experiment
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Exercise 5
: Show that
Binomial(
k
,
n, p
)
→
Poisson(
k
,
a
)
Exercise 4
: Prove it
Slide2020Common Distributions and Densities
Slide21Probability – What is it Exactly? 21
There are at least two interpretations of probability:Degree of belief in, or plausibility of, a propositionExample: It will snow in Geneva on Friday Relative frequency of outcomes in an infinite sequence of
identically repeated
trials
Example
:
trials: proton-proton collisions at the LHC
outcome: the creation of a Higgs boson
Slide22Likelihood
Slide2323Likelihood – 1
The likelihood function is simply the probability, or probability density function (pdf), evaluated at the observed data.Example 1: Top quark discovery (D0, 1995) p(D|
d
) =
Poisson(
D
|
d
) probability to get a count D
p
(17
|
d
) = Poisson(
17
|
d
)
likelihood
of observation
D
= 17
Poisson(
D
|
d
) = exp(-
d
)
d
D
/
D
!
24Likelihood – 2
Example 2: Multiple counts Di with a fixedtotal count NThis is an example of
a
multi-
binned
likelihood
Slide2525
Likelihood – 3Example 3: Red shift and distance modulus measurements of N = 580 Type Ia supernovae
This is an example of
an
un-binned
likelihood
Slide2626Likelihood – 4
Example 4: Higgs to γγThe discovery of the neutral Higgs boson in the di-photon final state made use of an an un-binned likelihood,where x = di-photon masses m = mass of new particle
w
= width of resonance
s
= expected signal
b = expected background
f
s = signal model
f
b
= background model
Exercise
6
: Show that a
binned multi-Poisson
likelihood yields an
un-binned likelihood of
this form as the bin widths
go to zero
Slide2727Likelihood – 5
Given the likelihood function we can answer questions such as:How do I estimate a parameter?How do I quantify its accuracy?How do I test an hypothesis?How do I quantify the significance of a result?Writing down the likelihood function requires:Identifying all that is known, e.g., the observationsIdentifying all that is unknown, e.g., the parameters
Constructing a probability model
for both
Slide2828Likelihood – 6
Example: Top Quark Discovery (1995), D0 Results knowns: D = 17 events B = 3.8 ± 0.6 background events unknowns: b expected background count s expected signal count d = b + s expected event countNote: we are uncertain about unknowns, so 17 ± 4.1 is a statement about d,
not about the observed count
17
!
Slide29Likelihood – 7Probability:
Likelihood: where B = Q / k δB = √Q / k
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Slide3030Summary
Statistic A statistic is any function of potential observationsProbabilityProbability is an abstraction that must be interpreted LikelihoodThe likelihood is the probability (or probability density) of potential observations
evaluated at the observed data
Slide31TutorialsLocation:http://www.hep.fsu.edu/~harry/
ESHEP13Download tutorials.tar.gzand unpack tar zxvf tutorials.tar.gzNeed: Recent version of Root
linked with RooFit and TMVA
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