Supporting Edge AI services is one of the most exciting features of future mobile networks. These services involve the collection and processing of voluminous data streams, right at the network edge, so as to offer real-time and accurate inferences to users. However, their widespread deployment is hampered by the energy cost they induce to the network. To overcome this obstacle, we propose a Bayesian learning framework for jointly configuring the service and the Radio Access Network (RAN), aiming to minimize the total energy consumption while respecting desirable accuracy and latency thresholds. Using a fully-fledged prototype with a software-defined base station (BS) and a GPU-enabled edge server, we profile a state-of-the-art video analytics AI service and identify new performance trade-offs. Accordingly, we tailor the optimization framework to account for the network context, the user needs, and the service metrics. The efficacy of our proposal is verified in a series of experiments and comparisons with neural network-based benchmarks.