Improving and Generalizing the ABCD Method for DataDriven Analyses at the LHC Using Bayesian Inference
Core Concepts
The Bayesian framework can improve and generalize the ABCD method by exploiting the mutual information in the multidimensional data, using softassignments instead of hard cuts, and handling multiple backgrounds simultaneously.
Abstract
The paper proposes using a Bayesian mixture model framework to improve and generalize the ABCD method, a commonly used datadriven technique in High Energy Physics (HEP) analyses at the Large Hadron Collider (LHC).
The key highlights are:

The ABCD method is limited to two independent observables and hard cuts to define regions, while the Bayesian framework can handle an arbitrary number of observables and uses softassignments instead of hard cuts.

The Bayesian framework can exploit the mutual information in the multidimensional data at the eventbyevent level, whereas the ABCD method only uses the information in two observables.

The Bayesian framework can handle multiple backgrounds simultaneously, while the ABCD method is limited to signal and a single background.

The authors demonstrate the advantages of the Bayesian framework over the ABCD method using a toy problem inspired by diHiggs production searches at the LHC. They show that the Bayesian approach outperforms the ABCD method in scenarios with small signal fractions.

The Bayesian framework provides the posterior distribution of the signal fraction, which is the key quantity of interest, instead of just a point estimate as in the ABCD method.
Overall, the Bayesian mixture model approach is a promising technique to improve datadriven analyses at the LHC by better exploiting the available information in the data.
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Improvement and generalization of ABCD method with Bayesian inference
Stats
"The signal fraction is estimated through its corresponding posterior."
"To obtain the predicted number of signal events we can use a softassignment strategy where we compute for each event the probability p(zn = sxn, θMAP) and obtain the estimated number of signal events Spred = Σn p(zn = sxn, θMAP)."
Quotes
"Instead of regions there is prior knowledge on the distribution of the K components over each one of the D independent observables. Then the data and its mutual information at the eventbyevent level provides the information to infer and learn the posterior on the class fractions, and in particular the posterior probability distribution for the signal fraction in the sample."
"Signal and background can be mixed in different proportions in all phase space. There is no need of a control region."
Deeper Inquiries
How can the Bayesian framework be extended to handle dependent observables and relax the assumption of conditional independence?
The Bayesian framework can be extended to handle dependent observables by incorporating a more complex probabilistic model that explicitly accounts for the dependencies among the observables. One approach is to use graphical models, such as Bayesian networks, which allow for the representation of conditional dependencies through directed acyclic graphs. In this framework, each observable can be connected to others, indicating how the distribution of one observable influences another.
To relax the assumption of conditional independence, one can introduce latent variables that capture the underlying relationships between observables. For instance, a hierarchical model can be employed where the dependencies are modeled at different levels, allowing for shared parameters that govern the behavior of multiple observables. This approach enables the model to learn from the data more effectively by utilizing the mutual information present in the dataset, thus improving the estimation of signal and background distributions.
Additionally, one can apply techniques such as copulas, which allow for the modeling of joint distributions while maintaining the marginal distributions of each observable. By using copulas, one can construct a joint distribution that reflects the dependencies among observables without losing the flexibility of the individual distributions. This extension of the Bayesian framework enhances its applicability to more complex scenarios encountered in LHC analyses, where observables often exhibit intricate correlations.
What are the challenges in applying the Bayesian mixture model approach to a realistic LHC analysis with a larger number of backgrounds and more complex signal and background distributions?
Applying the Bayesian mixture model approach to realistic LHC analyses presents several challenges, particularly when dealing with a larger number of backgrounds and more complex signal and background distributions.
Model Complexity: As the number of backgrounds increases, the complexity of the mixture model also rises. Each background may have its own distinct distribution, requiring careful parameterization and modeling. This complexity can lead to difficulties in convergence during the inference process, as the model may struggle to accurately estimate the parameters for all components simultaneously.
Data Sparsity: In scenarios with many backgrounds, the available data may be insufficient to robustly estimate the parameters of each background distribution. This sparsity can result in high uncertainty in the parameter estimates, making it challenging to distinguish between signal and background events effectively.
Computational Demands: The computational resources required for Bayesian inference increase significantly with the complexity of the model. The need for advanced sampling techniques, such as Markov Chain Monte Carlo (MCMC) or Variational Inference, can become computationally intensive, especially when dealing with highdimensional parameter spaces.
Prior Sensitivity: The choice of priors in Bayesian analysis can heavily influence the results, particularly in cases with limited data. In realistic LHC analyses, where the true distributions may not be well understood, selecting appropriate priors becomes a critical challenge. Misspecified priors can lead to biased estimates and affect the overall reliability of the analysis.
Signal Contamination: In realistic scenarios, the presence of signal contamination in control regions can complicate the estimation of background distributions. The Bayesian mixture model must account for this contamination, which may require additional modeling assumptions or the introduction of more complex latent structures to accurately capture the relationships between signal and background events.
Addressing these challenges requires careful consideration of model design, robust statistical techniques, and potentially the integration of additional data sources or prior knowledge to improve the reliability of the Bayesian mixture model in LHC analyses.
Can the Bayesian framework be combined with other machine learning techniques, such as neural networks, to further improve the modeling of the signal and background distributions?
Yes, the Bayesian framework can be effectively combined with other machine learning techniques, including neural networks, to enhance the modeling of signal and background distributions. This integration can leverage the strengths of both approaches, resulting in more robust and flexible models.
Bayesian Neural Networks: One of the most direct ways to combine Bayesian methods with neural networks is through Bayesian Neural Networks (BNNs). In BNNs, the weights of the neural network are treated as random variables with prior distributions. This allows for the incorporation of uncertainty in the model parameters, enabling the network to provide probabilistic predictions. The Bayesian approach helps mitigate overfitting, especially in scenarios with limited data, by regularizing the model through the prior distributions.
Variational Inference: The use of variational inference techniques can facilitate the training of complex neural network architectures within a Bayesian framework. By approximating the posterior distribution of the network parameters, one can efficiently learn from data while capturing the uncertainty associated with the predictions. This is particularly useful in highenergy physics, where the underlying distributions may be complex and multimodal.
Generative Models: Combining Bayesian methods with generative models, such as Generative Adversarial Networks (GANs) or Variational Autoencoders (VAEs), can enhance the modeling of signal and background distributions. These models can learn to generate synthetic data that mimics the underlying distributions, allowing for better estimation of the signal and background characteristics. The Bayesian framework can be applied to these generative models to quantify uncertainty in the generated samples.
Feature Learning: Neural networks excel at feature extraction and representation learning. By integrating neural networks into the Bayesian framework, one can automatically learn relevant features from the data that improve the separation between signal and background. This can lead to more accurate modeling of the distributions, as the learned features may capture complex relationships that traditional methods might miss.
Ensemble Methods: The Bayesian framework can also be combined with ensemble learning techniques, where multiple models are trained and their predictions are aggregated. This approach can enhance the robustness of the predictions by averaging out biases and reducing variance, leading to improved performance in distinguishing between signal and background events.
In summary, the combination of the Bayesian framework with machine learning techniques, particularly neural networks, offers a powerful approach to modeling complex distributions in LHC analyses. This synergy can lead to improved sensitivity and accuracy in identifying new physics signals amidst challenging backgrounds.