Given the pressing need for assuring algorithmic transparency, Explainable AI (XAI) has emerged as one of the key areas of AI research. In this paper, we develop a novel Bayesian extension to the LIME framework, one of the most widely used approaches in XAI – which we call BayLIME. Compared to LIME, BayLIME exploits prior knowledge and Bayesian reasoning to improve both the consistency in repeated explanations of a single prediction and the robustness to kernel settings. BayLIME also exhibits better explanation fidelity than the state-of-the-art (LIME, SHAP and Grad- CAM) by its ability to integrate prior knowledge from, e.g., a variety of other XAI techniques, as well as verification and validation (V&V) methods. We demonstrate the desirable properties of BayLIME through both theoretical analysis and extensive experiments. @en