Mikael Kuusela Statistical Issues in Unfolding Methods for High Energy Physics Master’s thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Technology in the Degree Programme in Engineering Physics and Mathematics. Espoo, July 26, 2012 Supervisor: Prof. Esko Valkeila Instructors: Prof. Victor Panaretos, D.Sc. (Tech.) Mikko Voutilainen Abstract of the Master’s Thesis Author: Mikael Kuusela Title: Statistical Issues in Unfolding Methods for High Energy Physics Supervisor: Prof. Esko Valkeila Instructors: Prof. Victor Panaretos, D.Sc. (Tech.) Mikko Voutilainen Degree Engineering Physics and Mathematics programme: Major subject: Mathematics Minor subject: Particle and Astrophysics Chair (code): Mat-1 Abstract: Due to the finite resolution of real-world particle detectors, any measurement conducted in experimental high energy physics is contaminated by stochastic smearing. This thesis studies the problem of unfolding these mea- surements to estimate the true physical distribution of the observable of interest before undesired detector effects. This problem is an ill-posed statistical inverse problem in the sense that straightforward inversion of the folding operator pro- duces in most cases highly oscillating unphysical solutions. The first contribution of this thesis is to provide a rigorous mathemati- cal understanding of the unfolding problem and the currently used unfolding techniques. To this end, we provide a mathematical model for the observations using indirectly observed Poisson point processes. We then explore the tools provided by both the frequentist and Bayesian paradigms of statistics for solv- ing the problem. We show that the main issue with regularized frequentist point estimates is that the bias of these estimators makes error estimation of the un- folded solution challenging. This problem can be resolved by using Bayesian credible intervals, but then one has to make an essentially arbitrary choice for the regularization strength of the Bayesian prior. Having gained a proper understanding about the issues involved in current unfolding methods, we proceed to propose a novel empirical Bayes unfolding technique. We solve the issue of choosing the spread of the regularizing Bayesian prior by finding a point estimate of the free hyperparameters via marginal maximum likelihood using a variant of the EM algorithm. This point estimate is then plugged into Bayes’ rule to summarize our understanding of the unknowns via the Bayesian posterior. We conclude with a computational demonstration of unfolding with a particular emphasis on empirical Bayes unfolding. Pages: vii+140 Language: English Date: July 26, 2012 Keywords: Unfolding, inverse problems, empirical Bayes, EM algorithm, Markov chain Monte Carlo, Poisson point processes, high energy physics i Diplomityön tiivistelmä Tekijä: Mikael Kuusela Työn nimi: Detektoriefektien poisto hiukkasfysiikan tilastollisessa data-analyysissä Työn valvoja: Prof. Esko Valkeila Työn ohjaajat: Prof. Victor Panaretos, TkT Mikko Voutilainen Koulutusohjelma: Teknillinen fysiikka ja matematiikka Pääaine: Matematiikka Sivuaine: Hiukkas- ja astrofysiikka Opetusyksikön (ent. professuuri) koodi: Mat-1 Tiivistelmä: Detektorien rajallisen resoluution takia jokainen kokeellisessa hiukkasfysiikassa tehtävä mittaus sisältää ei-toivottuja stokastisia efektejä. Tä- mä diplomityö käsittelee näiden detektoriefektien poistamista (engl. unfolding), millä tarkoitetaan kokeellisista efekteistä puhdistetun todellisen jakauman esti- moimista kiinnostuksen kohteena olevalle fysikaaliselle suureelle. Koska detekto- riefektejä kuvaavan operaattorin suora kääntäminen tuottaa useimmiten epäkel- poja oskilloivia ratkaisuja, kyseessä on haastava tilastollinen inversio-ongelma. Tämän työn ensimmäinen päämäärä on muodostaa tarkka matemaattinen malli detektoriefektien poistamiselle käyttäen epäsuorasti havaittuja Poisson- pisteprosesseja. Tämän jälkeen työssä analysoidaan sekä frekventistisen että bayesilaisen tilastotieteen näkökulmasta tehtävään käytettyjä nykymenetelmiä. Analyysi osoittaa, että frekventististen piste-estimaattorien tapauksessa löyde- tyn ratkaisun virherajojen estimointi on hankalaa johtuen regularisoitujen es- timaattorien harhaisuudesta. Ratkaisuksi ongelmaan on esitetty bayesilaisten luottamusvälien käyttöä, mutta tällöin herää kysymys siitä, kuinka regularisaa- tiovoimakkuutta säätelevä priorijakauma tulisi valita. Työssä esitetään näiden ongelmien ratkaisuksi uutta detektoriefektien pois- tomenetelmää, joka perustuu empiiriseen Bayes-estimointiin. Menetelmässä re- gularisoivan priorijakauman vapaat hyperparametrit estimoidaan suurimman reunauskottavuuden menetelmällä EM-algoritmia käyttäen, minkä jälkeen tämä piste-estimaatti sijoitetaan Bayesin kaavaan. Näin saatavaa posteriorijakaumaa voidaan sitten käyttää bayesilaisten luottamusvälien muodostamiseen. Tämän uuden detektoriefektien poistomenetelmän toiminta varmennetaan simulaatio- kokeita käyttäen. Sivumäärä: vii+140 Kieli: englanti Päivämäärä: 26.7.2012 Avainsanat: Detektoriefektien poisto, inversio-ongelma, empiirinen Bayes- estimointi, EM-algoritmi, Markovin ketju Monte Carlo, Poisson-pisteprosessi, hiukkasfysiikka ii Preface This work represents a collaboration between the Chair of Mathematical Statistics at École Polytechnique Fédérale de Lausanne (EPFL) and the CMS experiment at CERN, the European Organization for Nuclear Research, and was carried out in Switzerland during spring 2012. I would like to express my gratitude to my instructors Victor Panaretos and Mikko Voutilainen for their guidance and continuous support during this project, for answering the numerous questions I had and for providing feedback on the manuscript. I would in addition like to thank Esko Valkeila for supervising this thesis. It was also a great pleasure to work with the CMS Statistics Committee, and I would especially like to thank to Robert Cousins, Tommaso Dorigo and Louis Lyons for encouraging and enlightening discussions. I would also like to acknowl- edge the numerous CMS physicists who were willing to spend their time explaining unfolding to me in great detail. Furthermore, I would like to thank Yoav Zemel for useful comments on the manuscript, András László for interesting discussions and Otto Seiskari for allowing me to use this custom-made LATEX template. This thesis was funded by the CMS programme at Helsinki Institute of Physics and I would like to gratefully acknowledge their contribution to this project. Partial financial support was also provided by Aalto University and Aalto University Stu- dent Union. Thanks are also due to my two host organizations, EPFL and CERN, for providing office space and access to their facilities. Finally, I would like to thank my friends and family for all their support and encouragement in the course of this project. Espoo, July 26, 2012 Mikael Kuusela iii Contents Preface iii Notation and Abbreviations vi 1 Introduction 1 2 Formulation of the Unfolding Problem 6 2.1 Formulation as an Indirectly Observed Poisson Point Process . .6 2.1.1 Introduction to Point Processes . .6 2.1.2 Poisson Point Processes . .8 2.1.3 Indirectly Observed Poisson Point Processes . 11 2.1.4 Forward Model for Unfolding . 12 2.1.5 Discretization . 14 2.2 An Alternative Formulation . 17 3 Inference for Direct Observations 22 3.1 Maximum Likelihood Solution . 22 3.2 Frequentist Confidence Intervals . 23 3.3 Bayesian Credible Intervals . 24 3.4 Smoothing . 26 4 Frequentist Unfolding Techniques 28 4.1 Maximum Likelihood Estimation . 28 4.1.1 The Expectation-Maximization Algorithm . 32 4.1.2 Unfolding with the EM Algorithm . 34 4.2 Least Squares Estimation . 37 4.2.1 Truncated Singular Value Decomposition . 40 4.2.2 Tikhonov Regularization . 42 4.2.3 Error Estimation . 49 4.3 Choice of the Regularization Strength . 50 5 Bayesian Unfolding 54 5.1 Bayesian Inference for Unfolding . 54 5.2 Markov Chain Monte Carlo Sampling . 57 5.3 Prior Models . 61 iv 6 Empirical Bayes Unfolding 65 6.1 Parametric Empirical Bayes for Unfolding . 65 6.2 Marginal Maximum Likelihood Estimation with the MCEM Algorithm . 67 6.3 Empirical Bayes Unfolding with the Gaussian Smoothness Prior . 69 7 Computational Experiments 74 7.1 Gaussian Mixture Model . 74 7.1.1 Description of the Data . 74 7.1.2 Sampling Scheme . 76 7.1.3 Unfolding Results . 77 7.2 Inclusive Jet Cross Section . 88 7.2.1 Description of the Data . 88 7.2.2 Unfolding with Non-Uniform Binning . 92 7.2.3 Unfolding Results . 95 8 Discussion and Conclusions 106 8.1 Directions for Future Work . 106 8.2 Observations and Recommendations . 109 8.3 Concluding Remarks . 112 References 113 A Mathematical Background 117 A.1 Introduction to Probability Theory . 117 A.2 Statistical Inference . 126 A.3 Elements of Linear Algebra . 132 A.4 Inverse Problems . 137 v Notation and Abbreviations Notation Xi the random elements Xi are independent ?? 1A indicator function of the set A Ay Moore–Penrose pseudoinverse of the matrix A Ac complement of the set A bias(θ^) bias of the estimator θ^ of the parameter θ Bin(p; n) binomial distribution with n trials and success probability p cond(A) condition number of the matrix A δx Dirac measure at x det(A) determinant of the matrix A diag a1; : : : ; amin(m;n) m n diagonal matrix with diagonal elements m×n × a ; : : : ; a 1 min(m;n) f g convolution of f and g Γ(∗) gamma function ker(· A) kernel of the matrix A MSE[θ^] mean squared error of the estimator θ^ Mult(p; n) multinomial distribution with probabilities p and n trials ν Lebesgue measure