Practical Statistics for Particle Physics: Lecture Notes from CERN School HEP, Lecture notes of Particle Physics

Download Practical Statistics for Particle Physics: Lecture Notes from CERN School HEP and more Lecture notes Particle Physics in PDF only on Docsity!

Cosmology and Particle Physics

Kyle Cranmer,

New York University

Practical Statistics for Particle Physics

Cosmology and Particle Physics

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

‣ quantify our knowledge and uncertainty

‣ communicate results of experiments

Big questions:

‣ how do we make discoveries, measure or exclude theory parameters, etc.

‣ how do we get the most out of our data

‣ how do we incorporate uncertainties

‣ how do we make decisions

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

Introduction

Cosmology and Particle Physics

Other lectures

Fred James’s lectures Glen Cowan’s lectures Louis Lyons Bob Cousins gave a CMS lecture, may give it more publicly Gary Feldman “Journeys of an Accidental Statistician” The PhyStat conference series at PhyStat.org: 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=AT http://indico.cern.ch/conferenceDisplay.py?confId=a http://www.hepl.harvard.edu/~feldman/Journeys.pdf

Kyle Cranmer (NYU) Cosmology and Particle Physics CERN School HEP, Romania, Sept. 2011 Lecture 1 5

Kyle Cranmer (NYU) Cosmology and Particle Physics CERN School HEP, Romania, Sept. 2011 Preliminaries 7

Cosmology and Particle Physics

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

PDFs are always normalized to unity:

P (x ∈ [x, x + dx]) = f (x)dx

f (x)dx = 1

f (x)

x -3 -2 -1 0 1 2 3 f(x) 0

Cosmology and Particle Physics Parametric PDFs G(x|μ, σ) (μ,^ σ) Many familiar PDFs are considered parametric ‣ (^) eg. a Gaussian is parametrized by ‣ (^) defines a family of distributions ‣ (^) allows one to make inference about parameters I will represent PDFs graphically as below (directed acyclic graph) ‣ (^) every node is a real-valued function of the nodes below

Cosmology and Particle Physics Parametric PDFs G(x|μ, σ) (μ,^ σ)

G

x mu sigma

Many familiar PDFs are considered parametric ‣ (^) eg. a Gaussian is parametrized by ‣ (^) defines a family of distributions ‣ (^) allows one to make inference about parameters I will represent PDFs graphically as below (directed acyclic graph) ‣ (^) every node is a real-valued function of the nodes below

Cosmology and Particle Physics

The Likelihood Function

A Poisson distribution describes a discrete event count n for a real- valued mean_!. The likelihood of!_ given n is the same equation evaluated as a function of_!_ ‣ (^) Now it’s a continuous function ‣ (^) But it is not a pdf! Common to plot the -2 ln L ‣ (^) helps avoid thinking of it as a PDF ‣ (^) connection to!^2 distribution Likelihood-Ratio Interval example 68% C.L. likelihood-ratio interval for Poisson process with n= observed: !" ( μ ) = μ 3 exp(- μ )/3! Maximum at μ = 3. ∆ 2ln! = 1 2 for approximate ± 1 Gaussian standard deviation yields interval [1.58, 5.08] !"#$%&'(%)'+,'-)$."/.0''''''''''''' 1,'2,'345.,'67'789':;88<= L(μ) = P ois(n|μ) P ois(n|μ) = μ n e −μ n!

Cosmology and Particle Physics Change of variable x, change of parameter θ

• For pdf p(x| θ ) and change of variable from x to y(x):

p(y(x)| θ ) = p(x| θ ) / |dy/dx|.

Jacobian modifies probability density , guaranties that

P( y(x

1

)< y < y(x

2

) ) = P(x

1

< x < x

2

), i.e., that

Probabilities are invariant under change of variable x.

• – Mode of probabilityMode of probability densitydensity isis notnot invariant (so, e.g.,invariant (so, e.g.,

criterion of maximum probability density is ill-defined).

• Likelihood ratio is invariant under change of variable x.

(Jacobian in denominator cancels that in numerator).

• For likelihood! ( θ ) and reparametrization from θ to u( θ ):

! ( θ ) =! (u( θ )) (!).

• Likelihood! ( θ ) is invariant under reparametrization of

parameter θ (reinforcing fact that !" is not a pdf in θ ).

Bob Cousins, CMS, 2008 15

Cosmology and Particle Physics Different definitions of Probability 13 http://plato.stanford.edu/archives/sum2003/entries/probability-interpret/#3. |￿→ | ↑￿|

= 1 2 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 orthodox interpretation of probability in Quantum Mechanics. 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

Cosmology and Particle Physics

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:

Kolmogorov

axioms (1933)

Cosmology and Particle Physics

... in pictures (from Bob Cousins)

P, Conditional P, and Derivation of Bayes’ Theorem in Pictures A (^) B Whole space P(A) = P(B) = P(A B) = P(A|B) = P(B|A) = P(B) × P(A|B) = ×^ = P(A ∩ B) = P(A) × P(B|A) = ×^ = = P(A ∩ B) = P(A ∩ B) Bob Cousins, CMS, 2008!^ P(B|A) = P(A|B)^ ×^ P(B) / P(A) 7

Cosmology and Particle Physics

... in pictures (from Bob Cousins)

P, Conditional P, and Derivation of Bayes’ Theorem in Pictures A (^) B Whole space P(A) = P(B) = P(A B) = P(A|B) = P(B|A) = P(B) × P(A|B) = ×^ = P(A ∩ B) = P(A) × P(B|A) = ×^ = = P(A ∩ B) = P(A ∩ B) Bob Cousins, CMS, 2008!^ P(B|A) = P(A|B)^ ×^ P(B) / P(A) 7

Don’t forget about “Whole space”. I will drop it from the

notation typically, but occasionally it is important.

Ω