# Machine-Learning Application for a Likelihood Ratio Estimation Problem at LHC
## 3 Particle Physics Analyses Application
### YXH Fully-Hadronic Analysis
From the training set, 20% was used for validation, to validate the model and to monitor overfitting during the training phase. The DNN outputs event-level weights, assumed to be approximately independent of \(m_{H}\), that can be applied to an untagged region to produce the \(m_{Y}\) shape in the corresponding \(X_{bb}\)-tagged region. These weights are validated using data from the LSB. Figure 3 shows the impact of the reweighting on the distributions of several key analysis variables, using the two-prong merged LSB as an example region. Figure 2: Selection flow after and analysis regions of the \(Y\to XH\) search. The CRs (CR0, HSB0, HSB1), VRs (LSB0, LSB1), and SR are defined in the same way for all three analyses using the H candidate mass and \(D_{X_{bb}}\) score [6]. Three curves are shown in total, comparing the LSB0 data before and after DNN reweighting is applied to the target data distribution in LSB1. These variables are chosen to focus on kinematic variables over which the _background_ estimation is extrapolated to generate the SR prediction. Shape differences are observed after the application of the weights, and good agreement is observed for the reweighted shapes to the true tagged data in all distributions, suggesting a robust _background_ model. As the training is performed inclusively of the \(X\)-tagging, the same conclusion holds for the anomaly and two-prong resolved LSB regions. This approach enables a normalized distribution to be obtained and the _background_ normalization factor in SR is obtained by the fit procedure. Several sources of uncertainties are related to the method, i.e., effects that are not considered in the statistical model, but can affect the result of the measurement. They are called _systematic uncertainties_; their impact is taken into account by quantifying the corresponding variation on the _background_ shape. Three different kinds of such uncertainties were considered, as explained below. The first is the potential variation in the obtained weights due to differences in phase space between HSB (where the network is trained) and the Higgs mass window. The related uncertainty is calculated by obtaining an additional _background_ model, training the DNN in an alternate region of \(165\) GeV \(<m_{H}<200\) GeV. This region has approximately the same statistics and tagging efficiency as the nominal training region, helping to isolate the effect of the particular training region on the obtained output weights. Up and down variations are defined by symmetrizing the shape difference in \(m_{Y}\) between the two different models, creating an effect of \(\mathcal{O}(1-10)\%\) across the distribution. Figure 3: Distribution of the \(H\) candidate \(p_{T}\) (**a**) and \(\eta\) (**b**), along with the mass of \(J_{X}\) (**c**) and \(m_{Y}\) (**d**) in the merged LSB validation region, overlaying data from LSB1 with the data in LSB0 shown before (orange) and after (red) reweighting. The ratio of the LSB1 data to both the LSB0 data (orange) and the reweighted LSB0 data (red) is shown in the lower panel. Error bars indicate statistical uncertainties [6]. Another DNN variation is built to account for the finite statistics of the training sample and the random initialization of the weights. It is estimated with a bootstrap procedure [13], where a set of 100 bootstrap networks are trained, each time varying the training dataset by resampling it with a replacement. The correct way to evaluate this systematic is to use the event-level covariance matrix between all bootstrap weights, but, since it is computationally prohibitive, the interquartile range (IQR) for each event's weight distribution is considered as a good approximation of the uncertainty, along with the IQR of the normalization factor for each bootstrap training. Two additional templates are then defined with the median weight for each event, plus or minus half of the IQR, defining the upper and lower symmetric error bands. This corresponds to a \(\mathcal{O}(1)\%\) effect across \(m_{Y}\).