Deep mutational scanning (DMS) measures the effects of thousands of genetic variants in a protein simultaneously. The small sample size renders classical statistical methods ineffective. For example, p-values cannot be correctly calibrated when treating variants independently. We propose Rosace, a Bayesian framework for analyzing growth-based DMS data. Rosace leverages amino acid position information to increase power and control the false discovery rate by sharing information across parameters via shrinkage. We also developed Rosette for simulating the distributional properties of DMS. We show that Rosace is robust to the violation of model assumptions and is more powerful than existing tools.