Smart grid operation schemes can integrate prosumers by offering economic rewards in exchange for the desired response. In order to activate prosumers appropriately, such operation schemes require models of the dynamic uncertain price-response relationships. In this study, we combine the system identification of nonlinear dynamics with control (SINDyc) algorithm with Bayesian inference techniques based on Markov-chain Monte-Carlo sampling. We demonstrate this combination of two algorithms on an exemplary system in order to obtain parsimonious models alongside parameter uncertainty estimates. The precision of the identified models depends on the identification experiment and the parameterization of the algorithms. Such models may characterize the prosumer response and its uncertainty, thereby facilitating the integration of such entities into smart grid operation schemes.