High-resolution radar sensors are able to resolve multiple detections per object and therefore provide valuable information for vehicle environment perception. For instance, multiple detections allow to infer the size of an object or to more precisely measure the object's motion. Yet, the increased amount of data raises the demands on tracking modules: measurement models that are able to process multiple detections for an object are necessary and measurement-to-object associations become more complex. This paper presents a new variational radar model for tracking vehicles using radar detections and demonstrates how this model can be incorporated into a Random-Finite-Setbased multi-object filter. The measurement model is learned from actual data using variational Gaussian mixtures and avoids excessive manual engineering. In combination with the multiobject tracker, the entire process chain from the raw measurements to the resulting tracks is formulated probabilistically. The presented approach is evaluated on experimental data and it is demonstrated that the data-driven measurement model outperforms a manually designed model.