Phase randomizing processes in mesoscopic systems can be described in a phenomenological way within the Landauer-Büttiker formalism by attaching extra voltage probes to the sample. In this paper, it is shown that a perturbation treatment of this idea allows for the incorporation of such effects without the need of giving up the efficiency of recursive techniques commonly used for calculating the transmission coefficients. The technique is applied to a four-probe ring, where a Hall effect can be observed that originates from quantum interference rather than a Lorentz force acting on the elctrons. The influence of inelastic scattering on both the Hall resistance and the Aharonov-Bohm oscillations in the longitudinal resistance are examined.