With the deepening of neuroscience research, data mining of brain signals is becoming an emerging topic. Among various brain signals, electroencephalography (EEG) has attracted more and more attention due to its advantages of non-invasiveness, portability, and low cost. EEG modeling and analysis play a vital role in human healthcare. Although many machine learning algorithms have been successfully applied to data mining of EEG signals, few of them achieve a win-win in classification performance and interpretability. In this paper, we propose a Bayesian graph neural network named BayesEEGNet. Considering an electrical impulse between two nodes in the brain as a Poisson process, the countless electrical impulses generated by the brain in a period are represented as an infinite number of connection probability graphs. After coupling and transforming these probability graphs, we interpret the brain’s electrical activity state as the brain’s perceptual state. Benefiting from the joint optimization of Bayesian modules and deep neural networks, our model shows superior classification performance in sleep stage classification and emotion recognition tasks. Meanwhile, our model is able to learn interpretable functional connectivity relationships between EEG channels without any prior knowledge.