Knowing river discharge is vital for monitoring the fresh water cycle at the global scale. The Surface Water and Ocean Topography (SWOT) satellite mission will map river water surface elevations and inundated areas for rivers wider than one hundred meters worldwide. These observations can be used for estimating the river discharge. It is the global coverage which makes these observations particularly valuable, however in many cases there will be no data on the bathymetry and river bed properties available. That is why the problem of discharge estimation using solely the SWOT-type observations has received a noticeable attention recently.The attempts to solve this problem have expectedly confirmed that it is highly ill-posed and, therefore, additional data is useful. In particular, the use of the mean discharge estimates retrieved from the global scale hydrological databases (e.g. the Global Water Balance Model) have been accepted. However, taking into account the accuracy of such estimates and the issue of their relevance to the current time period, the problem is still posing a serious challenge. For example, in the results obtained by different methods reported up to date, the estimated hydrograph may suffer from a significant bias, which often makes a major contribution to the total estimation error.In this paper a new estimation method is suggested, which is specially designed to reduce the solution bias. The concept of this method is similar to the one of the Variational Expectation-Maximization method, however its perception and implementation are original and problem-oriented. In our method, the mean values of the unknown variables are obtained using the Bayesian estimator, whereas the 'shape' functions are updated using the variational data assimilation or generated directly using the inverted simplified hydraulic model. The two steps constitute an estimation cycle, which can be repeated after information exchange.The method has been validated using two available testing sets including 51 cases in total. It has demonstrated a much better robustness and reliability than the variational data assimilation method, and quite a promising performance in terms of accuracy. Since the proof of the concept was in the focus of this study, the issue of computational feasibility was not a priority. Nevertheless, the method in its current form can be applied at local/ regional or basin scales. For a possible global scale application, a generalized discharge estimator is suggested, where the major computational burden falls on the 'learning' stage, which is separated from the discharge prediction algorithm.