At the Wendelstein 7-X stellarator, the X-ray imaging crystal spectrometer provides line integrated measurements of ion and electron temperatures, plasma flows, as well as impurity densities from a spectroscopic analysis of tracer impurity radiation. In order to infer the actual profiles from line integrated data, a forward modeling approach has been developed within the Minerva Bayesian analysis framework. In this framework, the inversion is realized on the basis of a complete forward model of the diagnostic, including error propagation and utilizing Gaussian processes for generation and inference of arbitrary shaped plasma parameter profiles. For modeling of line integrated data as measured by the detector, the installation geometry of the spectrometer, imaging properties of the crystal, and Gaussian detection noise are considered. The inversion of line integrated data is achieved using the maximum posterior method for plasma parameter profile inference and a Markov chain Monte Carlo sampling of the posterior distribution for calculating uncertainties of the inference process. The inversion method shows a correct and reliable inference of temperature and impurity density profiles from synthesized data within the estimated uncertainties along the whole plasma radius. The application to measured data yields a good match of derived electron temperature profiles to data of the Thomson scattering diagnostic for central electron temperatures between 2 and 5 keV using argon impurities.