Deep neural networks have been increasingly used in various chemical fields. In the nature of a data-driven approach, their performance strongly depends on data used in training. Therefore, models developed in data-deficient situations can cause highly uncertain predictions, leading to vulnerable decision making. Here, we show that Bayesian inference enables more reliable prediction with quantitative uncertainty analysis. Decomposition of the predictive uncertainty into model- and data-driven uncertainties allows us to elucidate the source of errors for further improvements. For molecular applications, we devised a Bayesian graph convolutional network (GCN) and evaluated its performance for molecular property predictions. Our study on the classification problem of bio-activity and toxicity shows that the confidence of prediction can be quantified in terms of the predictive uncertainty, leading to more accurate virtual screening of drug candidates than standard GCNs. The result of log P prediction illustrates that data noise affects the data-driven uncertainty more significantly than the model-driven one. Based on this finding, we could identify artefacts that arose from quantum mechanical calculations in the Harvard Clean Energy Project dataset. Consequently, the Bayesian GCN is critical for molecular applications under data-deficient conditions.