7 Conclusions and Prospects

So Long, and Thanks for All the Fish.

– Douglas Adams

A large part of this thesis has dealt with the role of statistical learning techniques in the context of particle collider analyses, and their usefulness from a statistical inference perspective. After a broad introduction to the theoretical models of fundamental interactions and a summary of the main characteristics and working principles of the Compact Muon Solenoid (CMS) detector at the Large Hadron Collider (LHC), the fundamentals for statistical modelling at the LHC has been discussed. The relation between the theoretical parameters of interest and the experimental observations can only be modelled accurately by means of a complex simulation chain of the underlying physical processes and expected detector response. The generative-only nature of the simulation-based model combined with its high dimensionality make the definition of the probability density or likelihood function intractable, thus classical inference techniques cannot be applied to carry out statistical inference based on the acquired observations.

The statistical model for particle colliders can be described by a mixture model, each mixture component originating from a group of fundamental physical interactions. The latent variable structure of the generative model can be mapped to the different simulation steps in the simulation: process type, parton-level four-momenta, parton-shower outcome and detector readout. While the dimensionality of the latent space greatly increases for each subsequent step, the joint distribution can be factorised as a product of conditionals, the information about the parameters of interest being compactly expressed by the lowest dimensional latent variables. An efficient way to reduce the dimensionality of the data is thus to approximate the latent variables using the observations. This can be done by a well-calibrated combination of the different detector readouts, as is the case when using event reconstruction is performed, or by directly estimating the latent variables using supervised learning techniques trained on simulated observations.

Recent advances in supervised learning techniques have led to more accurate latent variable estimation that can scale to more data and use advanced non-linear transformations to obtain better performance in complex tasks, both in the context of classification and regression. Signal versus background probabilistic classification, a common conceptual framework for simplifying the event selection problem and constructing low-dimensional summaries in high-energy physics, has been formally proven to produce sufficient summary statistics for the mixture coefficients when the generative model is fully defined. The usefulness of probabilistic classification for such tasks, even in the optimal classifier case, cannot be guaranteed when nuisance parameters affect significantly the distribution of observed samples. In addition, particle identification and regression problems that augment the reconstruction output and can be tackled with machine learning techniques are also discussed. The use of deep learning techniques for advanced jet flavour tagging in CMS are used to exemplify the previous use case, which demonstrates the possible performance improvements due to the combined use of deep neural networks and non-standard input transformations that can deal with sequences. Newer machine learning methodologies that can deal with sets, graphs and other types of non-vector input coupled with powerful parallel hardware could be a promising path to substitute a larger part of the event reconstruction chain by latent variable approximations based on simulated observations, providing higher accuracy and throughput than hand-tuned algorithms.

An analysis using \(35.9\ \textrm{fb}^{-1}\) of data collected in 2016 by the CMS detector at the LHC was also included in this work. Proton-proton collisions at a centre-of-mass energy of 13 TeV were used to study the \(\textrm{pp} \rightarrow \textrm{HH} \rightarrow \textrm{b}\bar{\textrm{b}}\textrm{b}\bar{\textrm{b}}\) process in the context of the Standard Model (SM) and anomalous couplings effective field theory (EFT) extensions. The main challenge for this LHC analysis was the large background contribution from multi-jet QCD processes, so numerous that could not be modelled accurately by simulated observations. Hence, a data-driven estimation method, referred to as hemisphere mixing, was developed and validated on control regions to model the background contribution. The final summary statistic used in the analysis is based on the output of a probabilistic classifier, an ensemble of gradient boosted decision trees, trained using simulated signal observations and artificial events produced by the background estimation method. After assessing the different sources of systematic uncertainties and including their effect in the statistical model, a median expected limit obtained for SM HH production of \(419\ \textrm{fb}\) was obtained, which corresponds to approximately 37 times the SM expectation. The observed limit obtained is \(847\ \textrm{fb}\), which is about two standard deviations above the expected limit. Limits were also obtained for a set of EFT benchmarks, which summarise the kinematical properties of a large space of EFT models. The results of the combination of this analysis with other HH decay channels were also included. The estimation of QCD multijet backgrounds will likely remain an important issue for future jet-based analysis at the LHC, given that the biases of the data-driven estimation methods would become increasingly relevant as more data is available.

The ultimate goal of LHC analyses is statistical inference, in the form of hypothesis testing or parameter estimation. Machine learning techniques are useful to approximate latent variables which can then be used to construct powerful summary statistics for inference. In the presence of a generative model that depends on additional uncertain parameters, often referred to as nuisance parameters, the merits of classification or regression based summary statistics are greatly diminished. These concerns have motivated the development of a new family of techniques to construct powerful summary statistics that account directly for the final inference objective. By building and minimising loss functions that approximate the expected uncertainty on the parameters of interest, also accounting for the effect of nuisance parameters, the INFERNO approach can leverage recent machine learning technologies to construct better summary statistics for the inference problem at hand. These techniques were applied to a series of synthetic problems and were found to significantly outperform classification-based summary statistics (e.g. a deep neural network and the optimal classifier) when nuisance parameters are included in the problem. More experiments are needed to evaluate the value of this technique for real-word inference problems, such as those found in particle physics analyses.

As machine learning algorithms become increasingly popular in scientific contexts, it will be more important to formally describe the particularities of the problems we are trying to solve, in order to understand whether the tools at hand are answering the right questions. Otherwise we risk falling for the anti-pattern “if all you have is a hammer, everything looks like a nail”, which could significantly slow down the pace of scientific progress. This issue is particularly pressing for particle collider experiments, where the acquired familiarity with a given set of data analysis techniques might hinder the rigour in their application relative to the final objective. Some effort is then required to make sure of the role of a given tool is aligned with the task at hand instead on the subtleties of the tool itself. When using advanced statistical techniques or machine learning, the final analysis goal is of the upmost relevance and cannot be neglected in favour of procedural conventions. If those measures are coupled with open research practices and a careful use of domain-specific language and constructs in order to promote collaboration with other disciplines, better tools are likely to be developed which could in turn lead to major advancements in this research field.