A Bayesian model is based on a pair of probability distributions, known as the prior and sampling distributions. A wide range of fundamental machine learning tasks, including regression, classification, clustering, and many others, can all be seen as Bayesian models. We propose a new probabilistic programming abstraction, a typed Bayesian model, based on a pair of probabilistic expressions for the prior and sampling distributions. A sampler for a model is an algorithm to compute synthetic data from its sampling distribution, while a learner for a model is an algorithm for probabilistic inference on the model. Models, samplers, and learners form a generic programming pattern for model-based inference. They support the uniform expression of common tasks including model testing, and generic compositions such as mixture models, evidence-based model averaging, and mixtures of experts. A formal semantics supports reasoning about model equivalence and implementation correctness. By developing a series of examples and three learner implementations based on exact inference, factor graphs, and Markov chain Monte Carlo, we demonstrate the broad applicability of this new programming pattern.