The Bayesian Forecasting System is built of three processors: precipitation uncertainty processor (PUP), hydrologic uncertainty processor (HUP), and integrator. The PUP transforms a probabilistic quantitative precipitation forecast (PQPF) into a probability density function of the model river stage, given estimates of all other inputs; the model river stage is the output from a deterministic catchment model. The HUP characterizes two sources of uncertainty. A prior density quantifies the natural uncertainty about the actual river stage. A family of likelihood functions quantifies the uncertainty about the actual river stage due to imperfections of the catchment model and errors in the estimates of all inputs other than future precipitation. Then Bayes theorem yields the posterior density of the actual stage, conditional on the model stage, under the hypothesis that future precipitation is known. The densities from HUP and PUP are integrated into a predictive density that constitutes a probabilistic river stage forecast (PRSF). The theory underlying the HUP is outlined, and a prototype implementation of the HUP for the National Weather Service River Forecast System is described.