In order to create artificial enzymatic networks capable of increasingly complex behavior, an improved methodology in understanding and controlling the kinetics of these networks is needed. Here, we introduce a Bayesian analysis method allowing for the accurate inference of enzyme kinetic parameters and determination of most likely reaction mechanisms, by combining data from different experiments and network topologies in a single probabilistic analysis framework. This Bayesian approach explicitly allows us to continuously improve our parameter estimates and behavior predictions by iteratively adding new data to our models, while automatically taking into account uncertainties introduced by the experimental setups or the chemical processes in general. We demonstrate the potential of this approach by characterizing systems of enzymes compartmentalized in beads inside flow reactors. The methods we introduce here provide a new approach to the design of increasingly complex artificial enzymatic networks, making the design of such networks more efficient, and robust against the accumulation of experimental errors.