A key task in the emerging field of materials informatics is to use machine learning to predict a material's properties and functions. A fast and accurate predictive model allows researchers to more efficiently identify or construct a material with desirable properties. As in many fields, deep learning is one of the state-of-the art approaches, but fully training a deep learning model is not always feasible in materials informatics due to limitations on data availability, computational resources, and time. Accordingly, there is a critical need in the application of deep learning to materials informatics problems to develop efficient transfer learning algorithms. The Bayesian framework is natural for transfer learning because the model trained from the source data can be encoded in the prior distribution for the target task of interest. However, the Bayesian perspective on transfer learning is relatively unaccounted for in the literature, and is complicated for deep learning because the parameter space is large and the interpretations of individual parameters are unclear. Therefore, rather than subjective prior distributions for individual parameters, we propose a new Bayesian transfer learning approach based on the penalized complexity prior on the Kullback-Leibler divergence between the predictive models of the source and target tasks. We show via simulations that the proposed method outperforms other transfer learning methods across a variety of settings. The new method is then applied to a predictive materials science problem where we show improved precision for estimating the band gap of a material based on its structural properties.