Simultaneous $gamma$-ray measurements of gamma-ray burst (GRB) spectra and polarization offer a unique way to determine the underlying emission mechanism(s) in these objects as well as probing the particle acceleration mechanism(s) that lead to the observed $gamma$-ray emission. Herein we examine the jointly-observed data from POLAR and GBM of GRB 170114A to determine its spectral and polarization properties and seek to understand the emission processes that generate these observations. We aim to develop an extensible and statistically sound framework for these types of measurements applicable to other instruments. We leverage the existing 3ML analysis framework to develop a new analysis pipeline for simultaneously modeling the spectral and polarization data. We derive the proper Poisson likelihood for $gamma$-ray polarization measurements in the presence of background. The developed framework is publicly available for similar measurements with other $gamma$-ray polarimeters. The data are analyzed within a Bayesian probabilistic context and the spectral data from both instruments are simultaneously modeled with a physical, numerical synchrotron code. The spectral modeling of the data is consistent with a synchrotron photon model as has been found in a majority of similarly analyzed single-pulse GRBs. The polarization results reveal a slight trend of growing polarization in time reaching values of ~30% at the temporal peak of the emission. Additionally, it is observed that the polarization angle evolves with time throughout the emission. These results hint at a synchrotron origin of the emission but further observations of many GRBs are required to verify these evolutionary trends. Furthermore, we encourage the development of time-resolved polarization models for the prompt emission of GRBs as the current models are not predictive enough to enable a full modeling of our current data.
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