Scientific measurements often produce data that are mixtures of signal and one or more backgrounds. When class templates (for example, simulation-based priors) are biased or imprecise, they are usually treated as nuisance systematic uncertainties. In this work, we take a different approach: by embedding imprecise priors in a hierarchical Bayesian model and combining them with observed mixed data via Bayesian machine learning, we learn class shapes from the data that are closer to the true distributions than the original priors, and we obtain an unbiased posterior estimate of the signal fraction with calibrated uncertainties. That is, the method can improve simulations beyond simulation-based inference. We demonstrate the approach on a relevant collider-physics example: di-Higgs → 4b at the LHC. We also show how the same framework can improve the ABCD method. Additionally, we discuss how it can be used to learn distributions closer to the truth starting from leading-order (LO) priors, and how available next-to-leading-order (NLO) distributions could be used to assess that learning. The method, coined Template-Adapted Mixture Model (TAMM, arXiv:2604.022219), is straightforward to implement, scales to high-dimensional and complex distributions, and is broadly applicable in HEP, astrophysics, cosmology, and any domain where mixed data and imperfect priors occur.