Stronger symbolic summary statistics for the LHC

Nathalie Soybelman∗ Anja Butter Tilman Plehn
Institut für Theoretische Physik
Universität Heidelberg
Germany

Johann Brehmer
Center for Data Science
New York University
United States

Abstract
Analyzing the high-dimensional data collected at the Large Hadron Collider experiments often requires a balance between maximizing sensitivity and maintaining
interpretability by domain experts. We propose a new algorithm to constructpowerful summary statistics for LHC processes in the form of simple symbolic
expressions. First, we extract latent information from a chain of simulators; throughsymbolic regression on this data we then learn approximately sufficient statistics.
Observables constructed in this way can be used as plug-in replacements for established summary statistics, potentially improving the precision of scientific results
without adding any overhead. In Higgs production in weak boson fusion, our algorithm rediscovers well-known heuristics and proposes new, moderately complex
formulas that rival the new physics reach of neural networks.

1

Introduction

The experiments at the Large Hadron Collider (LHC) collect vast amounts of high-dimensionaldata, which need to be turned into precise measurements of the fundamental properties of nature.
While particle physicists have developed a chain of high-fidelity simulators for particle collisions, thelikelihood function that they implicitly define is intractable. This is a major challenge for inference
and a number of simulation-based (or likelihood-free) inference methods have been developed toovercome it [1–6].
Particle physicists typically tackle this problem by not working with the high-dimensional raw data
directly, but reducing its dimensionality by extracting a summary statistic. Once the data is onedimensional, the likelihood function can be estimated through univariate density estimation methods
like histograms, which makes both frequentist and Bayesian inference possible. However, designinga summary statistic is problem-specific and difficult. Traditionally, simple kinematic variables areoften chosen, but these are known to lose information, leading to less precise measurements [7–9].
Recently, neural summary statistics have been proposed that are approximately sufficient [10–12].
While they can be more powerful, they are less easily interpretable, sometimes perceived by moreconservative domain scientists as black boxes. In addition, integrating them in established data
analysis pipelines may require some effort. These (real and perceived) drawbacks have limited theiradaptation so far.
Here we introduce a new kind of summary statistics for LHC processes that is both approximately
optimal and highly interpretable. Our algorithm first extracts additional latent information from
LHC simulators and augments a training dataset with it. Next, we use symbolic regression on theaugmented data to learn the approximately sufficient statistics of the problem. The output of thisprocedure is a simple analytical expression like “pT 1 pT 2 sin ∆φ” that is self-explanatory to domainscientists, can be “stored” on a piece of paper, and can be directly plugged into a typical particle∗

nathalie@soybelman.de

35th Conference on Neural Information Processing Systems (NeurIPS 2021), Sydney, Australia.

physics analysis without adding overhead. We demonstrate the potential of this approach in ananalysis of the properties of the Higgs boson.

2

Related work

Simulation-based inference Many phenomena in science are modeled with simulators that do nothave a tractable likelihood function. Parameter inference in these models requires simulation-based
inference methods such as Approximate Bayesian Computation [2, 3] or techniques based on neuralsurrogates of the likelihood [13–16], likelihood ratio [10, 17–21], or posterior [22–28]; see Ref. [4]for a recent review and Refs. [6, 11, 12] for applications to particle physics.
Learned summary statistics Several methods have been proposed to learn powerful summary
statistics. This work is closely related to the S ALLY algorithm [10–12], in which a neural network (orboosted tree [29]) learns the score, the gradient of the log likelihood with respect to the parameters ofinterest. Neural summaries can also be learned by mutual information maximization [30] or based on
an exponential family approximation [31, 32]. Finally, it is possible to make summaries robust to
nuisance parameters [33, 34].
Even before these neural network–based methods, particle physicists have developed domain-specifictechniques for optimal summary statistics known as “optimal observables” [35–38]. Unlike ourmethod, these algorithms require approximations on the underlying physics process, for instance thedetector interactions.
Interpretable machine learning for particle physics In the context of particle physics, interpretable and explainable machine learning are still under-explored topics. References [39, 40]
proposed a method to extract a set of the most relevant high-level observables from a trained neuralnetwork. Symbolic regression has previously been used to learn scientific laws from data [41–44],
but also for interpretable classifiers in particle physics [45].

3

Optimal observables from symbolic regression

Setup We consider an LHC process in which we want to infer some parameters of interest θ fromobserved data {xi }. These data consist of multiple i. i. d. events, each characterized by a vector ofbasic particle properties such as reconstructed energies and masses. The relation between parametersand collected data is modeled by a simulator, which allows us to sample x ∼ p(x|θ), but the likelihood
function p(x|θ) is intractable. Our goal is to find a summary statistic t̂(x) such that inference on asummary dataset {t̂(xi )} allows a precise measurement of θ.
We focus on the case where the parameters θ are in the neighborhood of some reference parameter
point θ0 . This is common in precision measurements of the LHC, for instance when θ are the Wilson
coefficients of the Standard Model (SM) effective field theory (EFT) and the reference point is theSM, θ0 = 0.
Sufficient statistics In the local approximation |θ − θ0 |  1, the likelihood function is in the
exponential family and the components of the scoret(x|θ0 ) = ∇θ log p(x|θ)

θ0

(1)

are its sufficient statistics [10, 31]. In other words, as long as we consider parameter points close
to the reference point θ0 , the score represents the perfect summary statistic or optimal observables;
reducing a dataset to the values of the score for each event would not lose any information on θ.
Unfortunately, the score is in general intractable.
Learning the score Our goal is therefore to learn an approximation of the score. First, we leverage
the fact that particle physics simulators are not entirely black boxes, but let us access some informationthat characterizes the latent process in the simulator [11, 12]. More precisely, for each simulated
event x we can also access the ground-truth parton-level phase space point z as well as the squared2matrix element |M(z|θ)|2 . With these quantities, we can compute the joint score in the space oflatents and observablest(x, z|θ0 ) =∇θ σtot (θ0 )∇θ |M(z|θ0 )|2−|M(z|θ0 )|2
σtot (θ0 )(2)for each simulated event. Here σtot (θ) is the total cross section of the process.
The usefulness of the joint score is not immediately obvious, as it explicitly depends on the latent z,
which is only available in simulations. However, Refs. [10–12] have shown that the score t(x|θ0 )can be linked to the joint score t(x, z|θ0 ) through the minimum of the mean-squared-error (MSE)
functional:2t(x|θ0 ) = arg min Ex,z∼p(x,z|θ0 ) |g(x) − t(x, z|θ0 )| .
(3)
g(x)

We implement this equation in a practical algorithm with symbolic regression. Instead of the abstract
variational family g(x), we consider the function family of analytical expressions of a maximum
length, consisting of kinematic variables, mathematical operations, and numerical constants. Insteadof the exact minimization in Eq. (3), we use a genetic algorithm that performs random mutations on
a set of symbolic expressions and keeps solutions with a higher probability if they lower the MSE.
Hyperparameters allow us to choose the trade-off between complexity of the symbolic expression andthe value of the MSE in Eq. (3), which we expect to be related to the quality of the summary statistic.
All in all, our algorithm thus consists of two phases: first, we simulate events; for each event we savethe observable data x together with the joint score t(x, z). Second, we run a symbolic regression
algorithm on the joint score t(x, z) to learn an analytical expression t̂(x) that approximates the score.
The output of our algorithm is a simple symbolic expression like pT 1 pT 2 sin ∆φ. As an approximation of the score, summary statistics constructed in this way will be approximately equal to thesufficient statistics. They are therefore likely to improve the precision of the analysis results comparedto established heuristics. At the same time, they are as simple to handle as conventional kinematic
variables: these observables can simply be “stored” by writing their definition on a piece of paper.
Using it in a data analysis pipeline is equally straightforward and does not require, for instance, theevaluation of a neural network. Finally, the score can easily be interpreted by domain experts.
Implementation We simulate events with a standard simulator chain consisting of M AD G RAPH5 2.6.7 [46], P YTHIA 8.3 [47], and D ELPHES 3.4.2 [48]. M AD M INER 0.7.5 [49] is used
for book keeping and to compute the joint score. We use a modified version of P Y SR [50] to learn
the functional form of the score and add a parameter-fit afterburner to finetune the constants in theanalytical expression.

4

Experiments

We demonstrate the potential of our symbolically regressed observables in two precision measurements of the properties of the Higgs boson at the LHC. Signs of new physics are expected to manifest
themselves in such measurements as deviations from the Standard Model predictions.
First, we verify our algorithm in a toy problem and consider ZH production without decays, shower,and detector effects. This simple process has only two particles in the final state and is fully describedby only two degrees of freedom. We consider the problem of measuring the Wilson coefficientof the dimension-six operator OB [51], using the SM θ0 = 0 as reference point. As summary
statistic, our algorithm learns a polynomial in the transverse momentum pT and the difference of thepseudorapidities2 |ηZ − ηH |. This is in agreement with our expectations about the structure of theprocess.
We then focus on a realistic scenario: Higgs production in the weak boson fusion mode, including
a decay of the Higgs boson into muons and detector effects. This process has substantially more
complex kinematics and its final state involves two jets, collimated sprays of charged particles, inaddition to the muons that the Higgs decays into. Our goal is to measure the Wilson coefficient of the
CP -violating dimension-six operator OW W
f [51]. Again, we use the SM as reference point, θ0 = 0.
2

The pseudorapidity measures the angle between the direction of a particle and the beam axis.

3

1.0
pT1pT210 1
sin

pT1pT2
0.6
0.4

sin

68% CL

95% CL
1.0

0.5

SR: Eq. (4)SR: Eq. (5)10 2

SR: Eq. (4)SR: Eq. (5)
Sally

0.2
0.0

MSE in Eq. (3)Expected p-value

0.8

0.0
0.5
Theory parameter

1.0

Sally
0.3 0.4 0.5 0.6 0.7 0.8
Expected limit on at 68 % CL

Figure 1: Left: Expected p-values on θ based on different summary statistics, using a synthetic
dataset generated with θtrue = 0. We compare simple heuristics, more complex symbolic summary
statistics found by our symbolic regression (SR) algorithm, and a neural network trained with the
S ALLY algorithm. Right: Relation between the expected upper bounds with the MSE in Eq. (3).
Our symbolic regression algorithm finds a number of summary statistics of increasing complexity.
The simplest one is t̂1 (x) = sin ∆φ, where ∆φ is the difference between the azimuthal angles of thetwo jets. This is in fact an established heuristic [52]: When set to focus on simple expression, ouralgorithm thus rediscovers a quantity well-known to (and hand-engineered by) domain scientists.
At higher complexity, our algorithm learnst̂2 (x|θ0 ) = pT 1 pT 2 sin ∆φ(4)

and ultimatelyt̂3 (x|θ0 ) = −pT 1 (pT 2 + c)(a − b∆η) sin(∆φ + d) ,(5)
where pT i are the transverse momentum of the jets, ∆η is the difference in pseudorapidities between
the two jets and a, b, c, d are numerical constants.
We then study how powerful these summary statistics are by computing expected confidence regions
on the parameter of interest θ, using a synthetic “observed” dataset generated for θtrue = 0. The left
panel of Fig. 1 shows the expected p-values. The more complex summary statistics found by our
symbolic regression algorithm lead to stronger exclusion bounds on the parameter of interest than thesimple heuristics pT 1 pT 2 and sin ∆φ. In fact, our symbolic observables lead to confidence limits
virtually indistinguishable from those based on neural networks trained with the S ALLY algorithm.
Finally, in the right panel of Fig. 1 we investigate how predictive the MSE in Eq. (3) is of theperformance on the downstream task of measuring θ. Indeed, summary statistics that have a lower
MSE with respect to the joint score later enable more powerful measurements, visible as lower upperbounds on θ. However, the returns are diminishing: while more and more complex summary statistics
can further improve the fit to the joint score, this does not translate to substantial improvements in theparameter measurement. Moderately complex symbolic expressions provide a good compromise ofnear-perfect measurements and straightforward interpretability.

5

Outlook

Machine learning has a huge potential to improve the sensitivity of LHC measurements. However,the lack of interpretability of many models has been a limiting factor in the widespread adaptation of
new methods. We propose a new class of summary statistics based on symbolic regression. They
approximate the sufficient statistics and are therefore nearly optimal for a given problem. At the sametime, they are as easy to interpret, store, and implement in analyses as traditional kinematic heuristics.
For the theoretically interesting case of Higgs production in weak boson fusion, our algorithm
reinvents the well-known sine of the azimuthal angle between the tagging jets as a powerful summary
4

statistics. It also discovers new, slightly more complex expressions that allow us to measure fundamental properties of nature with higher precision, with almost no loss of performance compared toneural network–based summary statistics.
Acknowledgments
We are very grateful to Miles Cranmer for help on P Y SR, Kyle Cranmer for his insights, and Markus
Schumacher for enlightening and fun discussion on optimal observables and how to use them in Higgs
physics. The research of AB and TP is supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under grant 396021762 – TRR 257 Particle Physics Phenomenology
after the Higgs Discovery.

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8

A

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes](b) Did you describe the limitations of your work? [Yes] We describe the approximations
that our approach is based on.
(c) Did you discuss any potential negative societal impacts of your work? [No] Our work
is most relevant to research into the fundamental properties of nature, which we deem
a positive societal impact in itself. Due to their extremely general nature, it is however
possible that simulation-based inference methods can be applied to more nefarious
purposes.
(d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes]2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A] Our work
does not have new theoretical results. We cite the theoretical results we are building on.
(b) Did you include complete proofs of all theoretical results? [N/A] Our work does nothave new theoretical results. We cite the theoretical results we are building on.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No](b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [No] Not in this workshop paper, though we will make a longer version
with all relevant practical details available.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No](d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [No]4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes](b) Did you mention the license of the assets? [No] The licenses can be found through thecitations.
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information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable? [N/A](b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable? [N/A](c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation? [N/A]9