Traditional cloud-centric architectures for Internet-of-Things applications are being replaced by distributed approaches. The Edge and Fog computing paradigms crystallize the concept of moving computation towards the edge of the network, closer to where the data originates. This has important benefits in terms of energy efficiency, network load optimization and latency control. The combination of these paradigms with embedded artificial intelligence in edge devices, or Edge AI, enables further improvements. In turn, the development of blockchain technology and distributed architectures for peer-to-peer communication and trade allows for higher levels of security. This can have a significant impact on data-sensitive and mission-critical applications in the IoT. In this paper, we discuss the potential of an Edge AI capable system architecture for the Blockchain of Things. We show how this architecture can be utilized in health monitoring applications. Furthermore, by analyzing raw data directly at the edge layer, we inherently avoid the possibility of breaches of sensitive information, as raw data is never stored nor transferred outside of the local network