Single-pixel imaging (SPI) has the advantages of high-speed acquisition over a broad wavelength range and system compactness. Deep learning (DL) is a powerful tool that can achieve higher image quality than conventional reconstruction approaches. Here, we propose a Bayesian convolutional neural network (BCNN) to approximate the uncertainty of the DL predictions in SPI. Each pixel in the predicted image represents a probability distribution rather than an image intensity value, indicating the uncertainty of the prediction. We show that the BCNN uncertainty predictions are correlated to the reconstruction errors. When the BCNN is trained and used in practical applications where the ground truths are unknown, the level of the predicted uncertainty can help to determine whether system, data, or network adjustments are needed. Overall, the proposed BCNN can provide a reliable tool to indicate the confidence levels of DL predictions as well as the quality of the model and dataset for many applications of SPI.