The Tropical Rainfall Measuring Mission (TRMM)'s ‘day-1’ combined radar/radiometer algorithm uses a rain-profiling approach which gives as much importance to the measurements of the TRMM satellite's precipitation radar (PR) and the TRMM microwave imager (TMI) as their respective intrinsic ambiguities warrant, which avoids any ad hoc shortcuts that might introduce large biases in the rain estimates, yet which is simple enough to be operational when TRMM is launched in 1997. The algorithm is based on the idea of estimating the rain profile using the radar reflectivities, while constraining this inversion to be consistent with the radiometer-derived estimate of the total attenuation. To perform the data fusion, the problem is expressed in terms of drop-size-distribution variables. Starting with an a priori probability density function (pdf) for these variables, a Bayesian approach is used to condition the pdf successively on the radar and the radiometer measurements. The resulting algorithm is mathematically consistent and physically reasonable. The conditional variances which it calculates serve to quantify the accuracy of its estimates: small variances indicate that the TRMM observations can indeed be explained by the models used; large variances imply that the models are not sufficiently consistent with the measurements.