# Machine-Learning Application for a Likelihood Ratio Estimation Problem at LHC
## 3 Particle Physics Analyses Application
### YXH Fully-Hadronic Analysis
For all _signal_ regions, a cut is applied to the Higgs boson candidate \(D_{X_{bb}}\), along with a mass window requirement of \(75\;\mathrm{GeV}<m_{H}<145\;\mathrm{GeV}\). The further classification of events, where the Higgs boson mass candidate passes the \(D_{X_{bb}}\) selection and has mass between 145 and 200 GeV defines the _high-side band_ HSB1; HSB0 is in the same mass window, with the cut on \(D_{X_{bb}}\) reverted. Validation is performed in the _low-side band_ LSB, where the reconstructed Higgs boson mass is required to be between 65 and 75 GeV. LSB0 and LSB1 are similarly defined as having a Higgs boson candidate that fails or passes the \(D_{X_{bb}}\) tagging criterion, respectively. CR0 is defined as the set of events in which the \(J_{H}\) is in the SR mass window but fails the \(D_{X_{bb}}\) tagging. A scheme for the selection flow and the analysis region used for the _background_ estimation is shown in Figure 2.  
Background Estimation  
The overwhelming _background_ to the \(Y\to XH\) signal comprises high-\(p_{T}\) multi-jet events. Such processes are known to produce mismodelings in Monte Carlo simulations, making simulation-based _background_ estimation very challenging. Therefore, this analysis relies on a fully data-driven estimation of the _background_ in the SR. The shape of the expected \(m_{Y}\) distribution in the SR is obtained from data in the CR0 and weights are derived that can be applied to HSB0 to reproduce the shape found in HSB1. The baseline for this method is the verified assumption of independence of the \(D_{X_{bb}}\) cut efficiency from \(m_{H}\), in such a way that it is possible to define the reweighting function in a certain mass window and then apply it in another one. The reweighting function is defined as the ratio of the multi-dimensional probability distribution functions (PDFs) of the data in HSB1 to data in HSB0. In this analysis, the statistical procedure of direct importance estimation explained in Sections 2 and 2.1 is utilized, where the ratio is estimated directly from data. It is implemented via the training of a DNN, where the loss function (in Equation (5)) is minimized to produce weights that can accurately reproduce the observed ratio in data. The DNN is built using a fully connected sequential model from Keras with three inner layers, each with 20 neurons and a rectified linear unit (ReLU) activation function. To reduce the problem of overfitting during training, 10% of the connections among inner layers are randomly truncated (\"dropout\"). The last layer has a single output with a simple linear activation function. The model is trained using the Adam optimizer in Keras with Tensorflow as the backend. Training is performed using a batch size equal to the full dataset size for 1600 epochs, with early stopping if the value of the loss calculated on the validation dataset does not decrease for 100 subsequent epochs. Events are considered for training if they pass the analysis preselection, satisfy \(145~{}\mathrm{GeV}<m_{H}<175~{}\mathrm{GeV}\) and, additionally, have at least two track jets associated to the Higgs boson candidate. They are modeled as an unordered set of variables, namely the transverse momentum (\(p_{T}\)), the pseudorapidity (\(\eta\)), the azimuthal angle \(\phi\) and the energy of the Higgs boson candidate (E), the number of tracks associated to the Higgs boson candidate, the transverse momentum, \(\eta\), \(\phi\) and the mass of the first two track jets associated with the Higgs boson candidate, ordered in \(p_{T}\). Each variable \(x\) is standardized with the transformation \(x=\frac{(x-\mu)}{\sigma}\), where \(\mu\) and \(\sigma\) are its mean and standard deviation, respectively. The training was performed in the HSB, using data both in the tagged (HSB1) and untagged regions (HSB0) before applying the SRs categorization. This inclusive training enables use of a single weights set for merged, resolved and anomaly SRs. The dataset was divided into training and test sets using 70% and 30% of the full training dataset, respectively.