We present a framework for characterizing Bayesian classification methods. This framework can be thought of as a spectrum of allowable dependence in a given probabilistic model with the Naive Bayes algorithm at the most restrictive end and the learning of full Bayesian networks at the most general extreme. While much work has been carried out along the two ends of this spectrum, there has been surprisingly little done along the middle. We analyze the assumptions made as one moves along this spectrum and show the tradeoffs between model accuracy and learning speed which become critical to consider in a variety of data mining domains. We then present a general induction algorithm that allows for traversal of this spectrum depending on the available computational power for carrying out induction and show its application in a number of domains with different properties.