Abstract.
Parameter estimation in HEP experiments often involves Monte Carlo simulation to model the experimental response function. A typical application are forward-folding likelihood analyses with re-weighting, or time-consuming minimization schemes with a new simulation set for each parameter value. Problematically, the finite size of such Monte Carlo samples carries intrinsic uncertainty that can lead to a substantial bias in parameter estimation if it is neglected and the sample size is small. We introduce a probabilistic treatment of this problem by replacing the usual likelihood functions with novel generalized probability distributions that incorporate the finite statistics via suitable marginalization. These new PDFs are analytic, and can be used to replace the Poisson, multinomial, and sample-based unbinned likelihoods, which covers many use cases in high-energy physics. In the limit of infinite statistics, they reduce to the respective standard probability distributions. In the general case of arbitrary Monte Carlo weights, the expressions involve the fourth Lauricella function \(F_{D}\), for which we find a new finite-sum representation in a certain parameter setting. The result also represents an exact form for Carlson’s Dirichlet average \(R_{n}\) with \(n > 0\), and thereby an efficient way to calculate the probability generating function of the Dirichlet-multinomial distribution, the extended divided difference of a monomial, or arbitrary moments of univariate B-splines. We demonstrate the bias reduction of our approach with a typical toy Monte Carlo problem, estimating the normalization of a peak in a falling energy spectrum, and compare the results with previously published methods from the literature.
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References
Louis Lyons, Contemp. Phys. 54, 1 (2013)
F. Halzen, Eur. Phys. J. C 46, 669 (2006)
Lyndon Evans, Philip Bryant, JINST 3, S08001 (2008)
Kurt Binder, Dieter W. Heermann, Monte Carlo simulation in statistical physics, in Graduate Texts in Physics (Springer, 2010)
Roger Barlow, Christine Beeston, Comput. Phys. Commun. 77, 219 (1993)
Dmitry Chirkin, Likelihood description for comparing data with simulation of limited statistics (2016) available on-line at http://inspirehep.net/record/1413017?ln=de
G. Bohm, G. Zech, Nucl. Instrum. Methods Phys. Res. Sect. A 691, 171 (2012)
G. Bohm, G. Zech, Nucl. Instrum. Methods Phys. Res. Sect. A 748, 1 (2014)
F. Beaujean, H.C. Eggers, W.E. Kerzendorf, Mon. Not. R. Astron. Soc. 477, 3425 (2018)
Ritu Aggarwal, Allen Caldwell, Eur. Phys. J. Plus 127, 24 (2012)
C.H. Sim, Stat. Prob. Lett. 15, 135 (1992)
F. Di Salvo, G. Lovison, Parametric inference on samples with random weights, unpublished (1998)
F. Di Salvo, Integral Transforms Spec. Funct. 19, 563 (2008)
P.G. Moschopoulos, Ann. Inst. Stat. Math. 37, 541 (1985)
Harold Exton, Multiple hypergeometric functions and applications (Horwood, 1976)
James M. Dickey, J. Am. Stat. Assoc. 78, 628 (1983)
Willard Miller, J. Math. Phys. 13, 1393 (1972)
Akio Hattori, Tosihusa Kimura, J. Math. Soc. Jpn. 26, 1 (1974)
B.C. Carlson, J. Math. Anal. Appl. 7, 452 (1963)
Bille Chandler Carlson, Special Functions of Applied Mathematics (Academic Press, 1977)
Edward Neuman, Patrick J. Van Fleet, J. Comput. Appl. Math. 53, 225 (1994)
N. Balakrishnan, Discrete Multivariate Distributions (Wiley Online Library, 1997)
Ping Zhou, J. Comput. Appl. Math. 236, 94 (2011)
Youneng Ma, Jinhua Yu, Yuanyuan Wang, J. Appl. Math. 2014, 895036 (2014)
J.D. Keckic Mitrinovic, S. Dragoslav, The Cauchy Method of Residues (Springer Netherlands, 1984)
B.C. Carlson, J. Approx. Theory 67, 311 (1991)
G.S. Monti, The shifted-scaled Dirichlet distribution in the simplex, Universitat de Girona, Departament d’Informàtica i Matemàtica Aplicada (2011)
Jieqing Tan, Ping Zhou, Adv. Comput. Math. 23, 333 (2005)
Norman Lloyd Johnson, Samuel Kotz, Urn models and their application
Harold Jeffreys, Proc. R. Soc. London A 186, 453 (1946)
M.G. Aartsen, M. Ackermann, J. Adams, J.A. Aguilar, M. Ahlers, M. Ahrens, D. Altmann, T. Anderson, C. Arguelles, T.C. Arlen et al., Astrophys. J. 796, 109 (2014)
Ilía Nikolaevich Bronshtein and Konstantin Adol’fovich Semendyayev. Handbook of Mathematics (Springer Science & Business Media, 2013)
Peter J.M. van Laarhoven, Ton A.C.M. Kalker, J. Comput. Appl. Math. 21, 369 (1988)
Georgy P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, in Mathematical Monographs, Vol. 59 (American Mathematical Soc., 1984)
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Glüsenkamp, T. Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data. Eur. Phys. J. Plus 133, 218 (2018). https://doi.org/10.1140/epjp/i2018-12042-x
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DOI: https://doi.org/10.1140/epjp/i2018-12042-x