A physically based methodology to incorporate passive-microwave observations in a "rain-profiling algorithm" is developed for space- or air-borne radars at frequencies exhibiting attenuation. The rain-profiling algorithm deploys the Hitschfeld-Bordan formulation for reflectivity attenuation correction. In this formulation, the reflectivity -hydrometeor content (or rainfall rate) and reflectivity-attenuation relationships are expressed as function of one variable in the drop size distribution parametrization, namely, the multiplicative factor in a normalized gamma distribution. The multiplicative factor parameter, mean cloud water content, and three parameters describing the precipitation phase are estimated in a bayesian framework. This involves the minimization of differences between the 10, 19, 37 and 85GHz brightness temperature values predicted by a plane-parallel multi-layer radiative transfer model and those observed by space or air-borne radiometers. A variational approach is devised to perform the minimization. The methodology is applied to coincident airborne brightness temperature and radar profile observations originating in the Kwajalein Experiment of the Tropical Rainfall Measuring Mission. Results are compared to those yielded by an alpha adjustment technique, which determines the multiplicative factor of the drop size distribution using an non-adaptive estimate of the path integrated attenuation based on the observed 10GHz brightness temperature. Comparisons indicate a better consistency with all observations for the estimates derived in the bayesian framework. Recommendations regarding the application of the method to TRMM satellite observations are formulated based on the findings of the study.