Recent developments in cognitive psychology suggest models for knowledge and learning that often fall outside the realm of standard test theory. This paper concerns probability-based inference in terms of such models. The essential idea is to define a space of "student models"--simplified characterizations of students' knowledge, skill, or strategies, indexed by variables that signify their key aspects. From theory and data, one posits probabilities for the ways that students with different configurations in this space will solve problems, answer questions, and so on. Then the machinery of probability theory allows one to reason from observations of a student's actions to likely values of parameters in a student model. An approach using Bayesian inference networks is outlined. Basic ideas of structure and computation in inference networks are discussed and illustrated with an example from the domain of mixed-number subtraction. Six tables and 11 figures illustrate the discussion. (Contains 41 references.) (Author/SLD)