The Annals of Applied Statistics

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

Mikael Kuusela and Victor M. Panaretos

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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.

Article information

Ann. Appl. Stat., Volume 9, Number 3 (2015), 1671-1705.

Received: January 2014
Revised: July 2015
First available in Project Euclid: 2 November 2015

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Poisson inverse problem high energy physics Large Hadron Collider Poisson process regularization bootstrap Monte Carlo EM Algorithm


Kuusela, Mikael; Panaretos, Victor M. Statistical unfolding of elementary particle spectra: Empirical Bayes estimation and bias-corrected uncertainty quantification. Ann. Appl. Stat. 9 (2015), no. 3, 1671--1705. doi:10.1214/15-AOAS857.

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Supplemental materials

  • Supplement to “Statistical unfolding of elementary particle spectra: Empirical Bayes estimation and bias-corrected uncertainty quantification”. The supplement provides a comparison of the empirical Bayes and hierarchical Bayes approaches to unfolding; additional simulations results complementing those of Section 5; and technical material on the convergence and mixing of the MCMC sampler and on the Gaussian approximation used in the coverage studies.