Statistical unfolding of elementary particle spectra: Empirical Bayes estimation and bias-corrected uncertainty quantification

We consider the high energy physics unfolding problem where the goal is to estimate the spectrum of elementary particles given observations distorted by the limited resolution of a particle detector. This important statistical inverse problem arising in data analysis at the Large Hadron Collider at CERN consists in estimating the intensity function of an indirectly observed Poisson point process. Unfolding typically proceeds in two steps: one first produces a regularized point estimate of the unknown intensity and then uses the variability of this estimator to form frequentist confidence intervals that quantify the uncertainty of the solution. In this paper, we propose forming the point estimate using empirical Bayes estimation which enables a data-driven choice of the regularization strength through marginal maximum likelihood estimation. Observing that neither Bayesian credible intervals nor standard bootstrap confidence intervals succeed in achieving good frequentist coverage in this problem due to the inherent bias of the regularized point estimate, we introduce an iteratively bias-corrected bootstrap technique for constructing improved confidence intervals. We show using simulations that this enables us to achieve nearly nominal frequentist coverage with only a modest increase in interval length. The proposed methodology is applied to unfolding the $Z$ boson invariant mass spectrum as measured in the CMS experiment at the Large Hadron Collider.Comment: Published at http://dx.doi.org/10.1214/15-AOAS857 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:1401.827

Detektoriefektien poisto hiukkasfysiikan tilastollisessa data-analyysissä

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 measurements 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 produces in most cases highly oscillating unphysical solutions. The first contribution of this thesis is to provide a rigorous mathematical 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 solving 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 unfolded 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 hyper-parameters 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.Detektorien rajallisen resoluution takia jokainen kokeellisessa hiukkasfysiikassa tehtävä mittaus sisältää ei-toivottuja stokastisia efektejä. Tämä diplomityö käsittelee näiden detektoriefektien poistamista (engl. unfolding), millä tarkoitetaan kokeellisista efekteistä puhdistetun todellisen jakauman estimoimista kiinnostuksen kohteena olevalle fysikaaliselle suureelle. Koska detektoriefektejä kuvaavan operaattorin suora kääntäminen tuottaa useimmiten epäkelpoja oskilloivia ratkaisuja, kyseessä on haastava tilastollinen inversio-ongelma. Tämän työn ensimmäinen päämäärä on muodostaa tarkka matemaattinen malli detektoriefektien poistamiselle käyttäen epäsuorasti havaittuja Poissonpisteprosesseja. Tämän jälkeen työssä analysoidaan sekä frekventistisen että bayesilaisen tilastotieteen näkökulmasta tehtävään käytettyjä nykymenetelmiä. Analyysi osoittaa, että frekventististen piste-estimaattorien tapauksessa löydetyn ratkaisun virherajojen estimointi on hankalaa johtuen regularisoitujen estimaattorien harhaisuudesta. Ratkaisuksi ongelmaan on esitetty bayesilaisten luottamusvälien käyttöä, mutta tällöin herää kysymys siitä, kuinka regularisaatiovoimakkuutta säätelevä priorijakauma tulisi valita. Työssä esitetään näiden ongelmien ratkaisuksi uutta detektoriefektien poistomenetelmää, joka perustuu empiiriseen Bayes-estimointiin. Menetelmässä regularisoivan 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 simulaatiokokeita käyttäen

Multivariate Techniques for Identifying Diffractive Interactions at the LHC

31 pages, 14 figures, 11 tablesClose to one half of the LHC events are expected to be due to elastic or inelastic diffractive scattering. Still, predictions based on extrapolations of experimental data at lower energies differ by large factors in estimating the relative rate of diffractive event categories at the LHC energies. By identifying diffractive events, detailed studies on proton structure can be carried out. The combined forward physics objects: rapidity gaps, forward multiplicity and transverse energy flows can be used to efficiently classify proton-proton collisions. Data samples recorded by the forward detectors, with a simple extension, will allow first estimates of the single diffractive (SD), double diffractive (DD), central diffractive (CD), and non-diffractive (ND) cross sections. The approach, which uses the measurement of inelastic activity in forward and central detector systems, is complementary to the detection and measurement of leading beam-like protons. In this investigation, three different multivariate analysis approaches are assessed in classifying forward physics processes at the LHC. It is shown that with gene expression programming, neural networks and support vector machines, diffraction can be efficiently identified within a large sample of simulated proton-proton scattering events. The event characteristics are visualized by using the self-organizing map algorithm.Peer reviewe

Uncertainty quantification in unfolding elementary particle spectra at the Large Hadron Collider

This thesis studies statistical inference in the high energy physics unfolding problem, which is an ill-posed inverse problem arising in data analysis at the Large Hadron Collider (LHC) at CERN. Any measurement made at the LHC is smeared by the finite resolution of the particle detectors and the goal in unfolding is to use these smeared measurements to make nonparametric inferences about the underlying particle spectrum. Mathematically the problem consists in inferring the intensity function of an indirectly observed Poisson point process. Rigorous uncertainty quantification of the unfolded spectrum is of central importance to particle physicists. The problem is typically solved by first forming a regularized point estimator in the unfolded space and then using the variability of this estimator to form frequentist confidence intervals. Such confidence intervals, however, underestimate the uncertainty, since they neglect the bias that is used to regularize the problem. We demonstrate that, as a result, conventional statistical techniques as well as the methods that are presently used at the LHC yield confidence intervals which may suffer from severe undercoverage in realistic unfolding scenarios. We propose two complementary ways of addressing this issue. The first approach applies to situations where the unfolded spectrum is expected to be a smooth function and consists in using an iterative bias-correction technique for debiasing the unfolded point estimator obtained using a roughness penalty. We demonstrate that basing the uncertainties on the variability of the bias-corrected point estimator provides significantly improved coverage with only a modest increase in the length of the confidence intervals, even when the amount of bias-correction is chosen in a data-driven way. We compare the iterative bias-correction to an alternative debiasing technique based on undersmoothing and find that, in several situations, bias-correction provides shorter confidence intervals than undersmoothing. The new methodology is applied to unfolding the Z boson invariant mass spectrum measured in the CMS experiment at the LHC. The second approach exploits the fact that a significant portion of LHC particle spectra are known to have a steeply falling shape. A physically justified way of regularizing such spectra is to impose shape constraints in the form of positivity, monotonicity and convexity. Moreover, when the shape constraints are applied to an unfolded confidence set, one can regularize the length of the confidence intervals without sacrificing coverage. More specifically, we form shape-constrained confidence intervals by considering all those spectra that satisfy the shape constraints and fit the smeared data within a given confidence level. This enables us to derive regularized unfolded uncertainties which have by construction guaranteed simultaneous finite-sample coverage, provided that the true spectrum satisfies the shape constraints. The uncertainties are conservative, but still usefully tight. The method is demonstrated using simulations designed to mimic unfolding the inclusive jet transverse momentum spectrum at the LHC

Posterior Uncertainty Estimation via a Monte Carlo Procedure Specialized for Data Assimilation

Through the Bayesian lens of data assimilation, uncertainty on model parameters is traditionally quantified through the posterior covariance matrix. However, in modern settings involving high-dimensional and computationally expensive forward models, posterior covariance knowledge must be relaxed to deterministic or stochastic approximations. In the carbon flux inversion literature, Chevallier et al. proposed a stochastic method capable of approximating posterior variances of linear functionals of the model parameters that is particularly well-suited for large-scale Earth-system data assimilation tasks. This note formalizes this algorithm and clarifies its properties. We provide a formal statement of the algorithm, demonstrate why it converges to the desired posterior variance quantity of interest, and provide additional uncertainty quantification allowing incorporation of the Monte Carlo sampling uncertainty into the method's Bayesian credible intervals. The methodology is demonstrated using toy simulations and a realistic carbon flux inversion observing system simulation experiment

Background Modeling for Double Higgs Boson Production: Density Ratios and Optimal Transport

We study the problem of data-driven background estimation, arising in the search of physics signals predicted by the Standard Model at the Large Hadron Collider. Our work is motivated by the search for the production of pairs of Higgs bosons decaying into four bottom quarks. A number of other physical processes, known as background, also share the same final state. The data arising in this problem is therefore a mixture of unlabeled background and signal events, and the primary aim of the analysis is to determine whether the proportion of unlabeled signal events is nonzero. A challenging but necessary first step is to estimate the distribution of background events. Past work in this area has determined regions of the space of collider events where signal is unlikely to appear, and where the background distribution is therefore identifiable. The background distribution can be estimated in these regions, and extrapolated into the region of primary interest using transfer learning of a multivariate classifier. We build upon this existing approach in two ways. On the one hand, we revisit this method by developing a powerful new classifier architecture tailored to collider data. On the other hand, we develop a new method for background estimation, based on the optimal transport problem, which relies on distinct modeling assumptions. These two methods can serve as powerful cross-checks for each other in particle physics analyses, due to the complementarity of their underlying assumptions. We compare their performance on simulated collider data