Deep learning has gained increasing attention in the field of optical metrology and demonstrated great potential in solving a variety of optical metrology tasks, such as fringe analysis and phase unwrapping. However, deep neural networks cannot always produce a provably correct solution, and the prediction error cannot be easily detected and evaluated unless the ground-truth is available. This issue is critical for optical metrology, as the reliability and repeatability of the measurement are of major importance for high-stakes scenarios. In this paper, for the first time to our knowledge, we demonstrate that a Bayesian convolutional neural network (BNN) can be trained to not only retrieve the phase from a single fringe pattern but also produce uncertainty maps depicting the pixel-wise confidence measure of the estimated phase. Experimental results show that the proposed BNN can quantify the reliability of phase predictions under conditions of various training dataset sizes and never-before-experienced inputs. Our work allows for making better decisions in deep learning solutions, paving a new way to reliable and practical learning-based optical metrology.