Modeling Multi-Variate Gaussian Distributions and Analysis of Higgs Boson Couplings with the ATLAS Detector

Software tools developed for the purpose of modeling CERN LHC $pp$ collision data to aid in its interpretation are presented. Some measurements are not adequately described by a Gaussian distribution; thus an interpretation assuming Gaussian uncertainties will inevitably introduce bias, necessitating analytical tools to recreate and evaluate non-Gaussian features. One example is the measurements of Higgs boson production rates in different decay channels, and the interpretation of these measurements. The ratios of data to Standard Model expectations ($\mu$) for five arbitrary signals were modeled by building five Poisson distributions with mixed signal contributions such that the measured values of $\mu$ are correlated. Algorithms were designed to recreate probability distribution functions of $\mu$ as multi-variate Gaussians, where the standard deviation ($\sigma$) and correlation coefficients ($\rho$) are parametrized. There was good success with modeling 1-D likelihood contours of $\mu$, and the multi-dimensional distributions were well modeled within 1-$\sigma$ but the model began to diverge after 2-$\sigma$ due to unmerited assumptions in developing $\rho$. Future plans to improve the algorithms and develop a user-friendly analysis package will also be discussed.