Abstract
Precision measurements at the LHC often require analyzing highdimensional event data for subtle kinematic signatures, which is challenging for established analysis methods. Recently, a powerful family of multivariate inference techniques that leverage both matrix element information and machine learning has been developed. This approach neither requires the reduction of highdimensional data to summary statistics nor any simplifications to the underlying physics or detector response. In this paper, we introduce MadMiner , a Python module that streamlines the steps involved in this procedure. Wrapping around MadGraph5_aMC and Pythia 8, it supports almost any physics process and model. To aid phenomenological studies, the tool also wraps around Delphes 3, though it is extendable to a full Geant4based detector simulation. We demonstrate the use of MadMiner in an example analysis of dimensionsix operators in ttH production, finding that the new techniques substantially increase the sensitivity to new physics.
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Notes
The issue of likelihoodfree inference, the inference techniques discussed here, and MadMiner just as well apply in a Bayesian setting, see for instance Ref. [56].
Note that this approach is similar in spirit to the Matrix Element Method, which also uses partonlevel likelihoods and aims to estimate \(r(x  \theta _0, \theta _1)\) by calculating approximate versions of the integral in Eq. (3). But unlike the Matrix Element Method, our machine learningbased approach supports realistic shower and detector simulations and can be evaluated very efficiently.
The Fisher information defines a metric on the parameter space, giving rise to the field of information geometry [9, 73, 74]. In that formalism, we can also define “global” distances measured along geodesics, which are equivalent to the expected log likelihood ratio even beyond the local approximation of small \(\Delta \theta\) [75].
Fundamentally, the presented inference techniques also support new physics effects that affect e. g. the probabilities of shower splittings, but this is currently not supported in MadMiner.
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Acknowledgements
We would like to thank Zubair Bhatti, Lukas Heinrich, Alexander Held, and Samuel Homiller for their important contributions to the development of MadMiner . We are grateful to Joakim Olsson for his help with the tth data generation. We also thank Pablo de Castro, Sally Dawson, Gilles Louppe, Olivier Mattelaer, Duccio Pappadopulo, Michael Peskin, Tilman Plehn, Josh Rudermann, and Leonora Vesterbacka for fruitful discussions. Last but not least, we are grateful to the authors and maintainers of many opensource software packages, including Delphes 3 [65], Docker [108], Jupyter notebooks [109], MadGraph5_aMC [63], Matplotlib [110], NumPy [98], pylhe [111], Pythia 8 [112], Python [113], PyTorch [85], REANA [93], scikithep [114], scikitlearn [115], uproot [116], and yadage [117]. This work was supported by the U.S. National Science Foundation (NSF) under the awards ACI1450310, OAC1836650, and OAC1841471. It was also supported through the NYU IT High Performance Computing resources, services, and staff expertise. JB and KC are grateful for the support of the Moore–Sloan data science environment at NYU. KC is also supported through the NSF grant PHY1505463, while FK is supported by NSF grant PHY1620638 and U. S. Department of Energy grant DEAC0276SF00515.
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Appendix: Frequently Asked Questions
Appendix: Frequently Asked Questions
Here, we collect questions that are asked often, hoping to avoid misconceptions:

Does the whole event history not change when I change parameters?
No. In probabilistic processes such as those at the LHC, any given event history is typically compatible with different values of the theory parameters, but might be more or less likely. With “event history” we mean the entire evolution of a simulated particle collision, ranging from the initialstate and finalstate elementary particles through the parton shower and detector interactions to observables. The joint likelihood ratio and joint score quantify how much more or less likely one particular such evolution of a simulated event becomes when the theory parameters are varied.

If the network is trained on partonlevel matrix element information, how does it learn about the effect of shower and detector?
It is true that the “labels” that the networks are trained on, the joint likelihood ratio and joint score, are based on partonlevel information. However, the inputs into the neural network are observables based on a full simulation chain, after parton shower, detector effects, and the reconstruction of observables. It was shown in Ref. [59,60,61] that the joint likelihood ratio and joint score are unbiased, but noisy, estimators of the true likelihood ratio and true score (including shower and detector effects). A network trained in the right way will, therefore, learn the effect of shower and detector. We illustrate this mechanism in Sect. 5.1 in a onedimensional problem.

Can this approach be used for signalbackground classification?
Yes. In the simplest case, where the signal and background hypothesis do not depend on any additional parameters, the Carl, Rolr, or Alice techniques can be used to learn the probability of an individual event being signal or background. If there are parameters of interest such as a signal strength or the mass of a resonance, the score becomes useful and techniques such as Sally, Rascal, Cascal, and Alices can be more powerful.
The techniques that use the joint likelihood ratio or score require less training data when the signal and background processes populate the same phasespace regions. If this is not the case, these methods still apply, but will not offer an advantage over the traditional training of binary classifiers.

What if the simulations do not describe the physics accurately?
No simulator is perfect, but many of the techniques used for incorporating systematic uncertainties from mismodeling in the case of multivariate classifiers can also be used in this setting. For instance, often, the effect of mismodeling can be corrected with simple scale factors and the residual uncertainty incorporated with nuisance parameters. MadMiner can handle such systematic uncertainties as discussed above. If only particular phasespace regions are problematic, for instance those with lowenergy jets, we recommend to exclude these parameter regions with suitable selection cuts. If the kinematic distributions are trusted, but the overall normalization is less well known, a datadriven normalization can be used.
Of course, there is no silver bullet, and if the simulation code is not trustworthy at all in a particular process and the uncertainty cannot be quantified with nuisance parameters, these methods (and many more traditional analysis methods) will not provide accurate results.

Is the neural network a black box?
Neural networks are often criticized for their lack of explainability. It is true that the internal structure of the network is not directly interpretable, but in MadMiner , the interpretation of what the network is trying to learn is clearly connected to the matrix element. In practical terms, one of the challenges is to verify whether a network has been successfully trained. For that purpose, many crosschecks and diagnostic tools are available to make sure that this is the case:

checking the loss function on a separate validation sample;

training of multiple network instances with independent random seeds, as discussed above;

checking the expectation values of the score and likelihood ratio against their known true values, see Ref. [61];

varying of the reference hypothesis in the likelihood ratio, see Ref. [61];

training classifiers between data reweighted with the estimated likelihood ratio and original data from a new parameter point, see Ref. [61];

validating the inference techniques in lowdimensional problems with histograms, see Sect. 5.1;

validating the inference techniques on a partonlevel scenario with tractable likelihood function, see Sect. 5.2; and

checking the asymptotic distribution of the likelihood ratio against Wilks’ theorem [69,70,71].
Finally, when limits are set based on the Neyman construction with toy experiments (rather than using the asymptotic properties of the likelihood ratio), there is a coverage guarantee: the exclusion contours constructed in this way will not exclude the true point more often than the confidence level. No matter how wrong the likelihood, likelihood ratio, or score function estimated by the neural network is, the final limits might lose statistical power, but will never be too optimistic.


Are you trying to replace PhD students with a machine?
As a preemptive safety measure against scientists being made redundant by automated inference algorithms, we have implemented a number of bugs in MadMiner . It will take skilled physicists to find them, ensuring safe jobs for a while. More seriously, just as MadGraph automated the process of generating events for an arbitrary hard scattering process, MadMiner aims to contribute to the automation of several steps in the inference chain. Both developments enhance the productivity of physicists.
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Brehmer, J., Kling, F., Espejo, I. et al. MadMiner: Machine LearningBased Inference for Particle Physics. Comput Softw Big Sci 4, 3 (2020). https://doi.org/10.1007/s4178102000352
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DOI: https://doi.org/10.1007/s4178102000352