  X-ray polarimetry has the potential to make key-contributions to our
understanding of galactic compact objects like binary black hole systems and
neutron stars, and extragalactic objects like active galactic nuclei, blazars,
and Gamma Ray Bursts. Furthermore, several particle astrophysics topics can be
addressed including uniquely sensitive tests of Lorentz invariance. In the
energy range from 10 keV to several MeV, Compton polarimeters achieve the best
performance. In this paper we evaluate the benefit that comes from using the
azimuthal and polar angles of the Compton scattered photons in the analysis,
rather than using the azimuthal scattering angles alone. We study the case of
an ideal Compton polarimeter and show that a Maximum Likelihood analysis which
uses the two scattering angles lowers the Minimum Detectable Polarization (MDP)
by ~20% compared to a standard analysis based on the azimuthal scattering
angles alone. The accuracies with which the polarization fraction and the
polarization direction can be measured improve by a similar amount. We conclude
by discussing potential applications of Maximum Likelihood analysis methods for
various polarimeter experiments.
