Transfer learning (TL) has emerged as a powerful tool to supplement data collected for a target task with data collected for a related source task. The Bayesian framework is natural for TL because information from the source data can be incorporated in the prior distribution for the target data analysis. In this paper, we propose and study Bayesian TL methods for the normal-means problem and multiple linear regression. We propose two classes of prior distributions. The first class assumes the difference in the parameters for the source and target tasks is sparse, i.e., many parameters are shared across tasks. The second assumes that none of the parameters are shared across tasks, but the differences are bounded in $\ell_2$-norm. For the sparse case, we propose a Bayes shrinkage estimator with theoretical guarantees under mild assumptions. The proposed methodology is tested on synthetic data and outperforms state-of-the-art TL methods. We then use this method to fine-tune the last layer of a neural network model to predict the molecular gap property in a material science application. We report improved performance compared to classical fine tuning and methods using only the target data.

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