The statistical community has brought logical rigor and mathematical precision to the problem of using data to make inferences about a model’s parameter values. The TETRAD project, and related work in computer science and statistics, aims to apply those standards to the problem of using data and background knowledge to make inferences about a model’s specification. We begin by drawing the analogy between parameter estimation and model specification search. We then describe how the specification of a structural equation model entails (...) familiar constraints on the covariance matrix for all admissible values of its parameters; we survey results on the equivalence of structural equation models, and we discuss search strategies for model specification. We end by presenting several algorithms that are implemented in the TETRAD II program. (shrink)