Speaker
Description
The likelihood ratio (LR) plays an important role in statistics and many domains of science. The Neyman-Pearson lemma states that it is the most powerful test statistic for simple statistical hypothesis testing problems [1] or binary classification problems. Likelihood ratios are also key to Monte Carlo importance sampling techniques [2]. Unfortunately, in many areas of study the probability densities comprising the likelihood ratio are defined by implicit models, and so are intractable to compute explicitly [3].
Neural based LR estimation using probabilistic classification has therefore had a significant impact in these domains, providing a scalable method for determining an intractable LR from simulated datasets via the so-called ratio trick [4, 5]. These approaches typically adhere to the standard Kolmogorov axioms of probability theory [6]. In particular, they assume the first axiom: the probability of an event is a non-negative real number. However, there are settings in which synthetically generated data (e.g. Monte Carlo sampling) $\{(x_{i}, w_{i})\}^{N}_{i=1}$ contains weights that are negative $w_{i} < 0$ [7, 8]. These negative weights are a symptom of a class of distribution known as quasiprobabilities, which do not adhere to the first Kolmogorov axiom. Consequently, the probabilistic-like distribution has a negative density [9]; $q(x) < 0$ for some x.
In high energy physics, negative weights/densities are a commonly observed feature of Monte Carlo simulated proton-proton (pp) collision datasets [10-13]. Whether it be due to quantum interference between Standard Model and new physics processes, or algorithms that match/merge matrix element calculations of beyond leading order Quantum Chromodynamic processes with parton showers, Monte Carlo simulation codes often introduce negatively weighted data.
This work will present a general approach to extending the neural based LR trick to quasiprobabilistic distributions. It will demonstrate that a new loss function, combined with signed probability measures (Hahn-Jordan decomposition), can be used to decompose the likelihoods into signed mixture models. A quasiprobabilistic analog of the Likelihood Ratio is then constructed using a ratio of signed mixture models. The technique is demonstrated using di-Higgs production via gluon-gluon fusion in $pp$ collisions at the Large Hadron Collider [14].
References:
[1] Jerzy Neyman, Egon Sharpe Pearson, and Karl Pearson. Ix. on the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 231(694-706):289–337, 1933.
[2] Christian Lemieux. Monte Carlo and Quasi-Monte Carlo Sampling. Springer, New York, NY, USA, 2009.
[3] Peter J. Diggle and Richard J. Gratton. Monte carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society: Series B (Methodological), 46(2):193–212, 1984.
[4] Masashi Sugiyama, Taiji Suzuki, and Takafumi Kanamori. Density Ratio Estimation in Machine Learning. Cambridge University Press, 2012.
[5] Kyle Cranmer, Juan Pavez, and Gilles Louppe. Approximating likelihood ratios with calibrated discriminative classifiers, 2016.
[6] A.N. Kolmogorov. Grundbegriffe der Wahrscheinlichkeitsrechnung. Number 1. Springer Berlin, Heidelberg, 1933.
[7] Stefano Frixione and Bryan R Webber. Matching nlo qcd computations and parton shower simulations. Journal of High Energy Physics, 2002(06):029–029, June 2002.
[8] Paolo Nason and Giovanni Ridolfi. A positive-weight next-to-leading-order monte carlo forzpair hadroproduction. Journal of High Energy Physics, 2006(08):077–077, August 2006.
[9] Richard Phillips Feynman. Negative probability. 1984.
[10] Lyndon Evans and Philip Bryant. Lhc machine. Journal of Instrumentation, 3(08):S08001, aug 2008.
[11] ATLAS collaboration. Modelling and computational improvements to the simulation of single vector-boson plus jet processes for the atlas experiment. Journal of High Energy Physics, 2022(8), August 2022.
[12] Andrea Valassi, Efe Yazgan, et al, 'Challenges in monte carlo event
generator software for high-luminosity LHC.' Computing and Software for Big Science, 5(1), May 2021.
[13] ATLAS. Study of ttbb and ttW background modelling for ttH analyses. 2022.
[14] Amos Breskin and Rudiger Voss. The CERN Large Hadron Collider: Accelerator and Experiments. CERN, Geneva, 2009.
