An analytical Bayesian inversion of the JET Interferometry line integrated densities into density profiles and associated uncertainty information, is demonstrated. These are used, with a detailed forward model of plasma polarimetry, to predict the rotation and ellipticity for the JET polarimeter. This includes the lateral channels, for over 45,000 time points over 1313 JET pulses. Good agreement with measured values is shown for a number of channels. For the remaining channels, the requirement of a more detailed model of the diagnostic is demonstrated. A commonly used approximation for the Cotton-Mouton effect on the lateral channels is also evaluated.