Most traditional collaborative filtering (CF) methods only use the user-item rating matrix to make recommendations, which usually suffer from cold-start and sparsity problems. To address these problems, on the one hand, some CF methods are proposed to incorporate auxiliary information such as user/item profiles; on the other hand, deep neural networks, which have powerful ability in learning effective representations, have achieved great success in recommender systems. However, these neural network based recommendation methods rarely consider the uncertainty of weights in the network and only obtain point estimates of the weights. Therefore, they maybe lack of calibrated probabilistic predictions and make overly confident decisions. To this end, we propose a new Bayesian dual neural network framework, named BDNet, to incorporate auxiliary information for recommendation. Specifically, we design two neural networks, one is to learn a common low dimensional space for users and items from the rating matrix, and another one is to project the attributes of users and items into another shared latent space. After that, the outputs of these two neural networks are combined to produce the final prediction. Furthermore, we introduce the uncertainty to all weights which are represented by probability distributions in our neural networks to make calibrated probabilistic predictions. Extensive experiments on real-world data sets are conducted to demonstrate the superiority of our model over various kinds of competitors.