We couple geomechanical modeling with seismic velocity to enhance the prediction of pressure and stresses in complex geologic settings. In these settings, pressure is controlled by mean and shear stresses rather than by only the vertical (overburden) stress. We estimate total mean and shear stresses from a geomechanical model. Effective mean and shear stresses are calculated from velocity using a relationship that we develop between velocity and these stresses. The pressure prediction process is iterated to attain convergence between the predicted pressure field and the one input in the geomechanical model. We also explicitly predict the full stress tensor. We apply our method along with the standard, vertical-effective-stress method to a salt basin beneath the Sigsbee Escarpment in the Mad Dog field, Gulf of Mexico. The methods are constrained to the same pressure data along a calibration well and are then used to predict pressure and stresses across the basin. We find that salt and basin bathymetry substantially perturb the stress field. The pressures predicted by the two methods differ the least at the calibration well and the most in areas where the total mean and shear stresses are the most different from those at the same burial depth at the calibration well. Our method is shown to predict pressures measured along a subsalt well better than the standard, vertical method. We calculate minimum stress and the drilling window along a vertical profile near salt and find that they significantly differ from the ones predicted by the standard, vertical method.