Personalized recommender systems are playing an increasingly important role for online consumption platforms. Because of the multitude of relationships existing in recommender systems, Graph Neural Networks (GNNs) based approaches have been proposed to better characterize the various relationships between a user and items while modeling a user’s preferences. Previous graph-based recommendation approaches process the observed user-item interaction graph as a ground-truth depiction of the relationships between users and items. However, especially in the implicit recommendation setting, all the unobserved user-item interactions are usually assumed to be negative samples. There are missing links that represent a user’s future actions. In addition, there may be spurious or misleading positive interactions. To alleviate the above issue, in this work, we take a first step to introduce a principled way to model the uncertainty in the user-item interaction graph using the Bayesian Graph Convolutional Neural Network framework. We discuss how inference can be performed under our framework and provide a concrete formulation using the Bayesian Probabilistic Ranking training loss. We demonstrate the effectiveness of our proposed framework on four benchmark recommendation datasets. The proposed method outperforms state-of-the-art graph-based recommendation models. Furthermore, we conducted an offline evaluation on one industrial large-scale dataset. It shows that our proposed method outperforms the baselines, with the potential gain being more significant for cold-start users. This illustrates the potential practical benefit in real-world recommender systems.