Evolutionary algorithms are by now well-known and appreciated in a number of disciplines including the emerging field of data mining. In the last couple of decades, Bayesian learning has also experienced enormous growth in the statistical literature. An interesting question refers to the possible synergetic effects between Bayesian and evolutionary ideas, particularly with an eye to large-sample applications. This chapter presents a new approach to classification based on the integration of a simple local Bayesian engine within the learning classifier system rulebased architecture. The new algorithm maintains and evolves a population of classification rules which individually learn to make better predictions on the basis of the data they get to observe. Certain reinforcement policy ensures that adequate teams of these learning rules be available in the population for every single input of interest. Links with related algorithms are established, and experimental results suggesting the parsimony, stability and usefulness of the approach are discussed.