We study the regulation of a monopolistic firm that provides a non-marketed output based on multiple substitutable inputs. The regulator is able to observe the effectiveness of the provision, but faces information asymmetries with respect to the efficiency of the firm's activities. Motivated by the example of electricity transmission services, we consider a setting where one input (grid expansion) and the output (uninterrupted electricity transmission) are observable, while another input (sophisticated grid operation) and related costs are not. Multi-dimensional information asymmetries are introduced by discrete distributions for the functional form of the marginal rate of substitution between the inputs as well as for the input costs. For this novel setting, we investigate the theoretically optimal Bayesian regulation mechanism. We find that the first best solution cannot be obtained in case of shadow costs of public funding. The second best solution implies separation of the most efficient type with first best input levels, and upwards distorted (potentially bunched) observable input levels for all other types. Moreover, we compare these results to a simpler non-Bayesian approach and hence, bridge the gap between the academic discussion and regulatory practice. We provide evidence that under certain conditions, a single contract non-Bayesian regulation can indeed get close to the second best of the Bayesian menu of contracts regulation.