Conventional methods for protein structure determination from NMR data rely on the ad hoc combination of physical forcefields and experimental data, along with heuristic determination of free parameters such as weight of experimental data relative to a physical forcefield. Recently, a theoretically rigorous approach was developed which treats structure determination as a problem of Bayesian inference. In this case, the forcefields are brought in as a prior distribution in the form of a Boltzmann factor. Due to high computational cost, the approach has been only sparsely applied in practice. Here, we demonstrate that the use of generative probabilistic models instead of physical forcefields in the Bayesian formalism is not only conceptually attractive, but also improves precision and efficiency. Our results open new vistas for the use of sophisticated probabilistic models of biomolecular structure in structure determination from experimental data.