This paper analyzes the problem of learning the structure of a Bayes net in the theoretical framework of Gold’s learning paradigm. Bayes nets are one of the most prominent formalisms for knowledge representation and probabilistic and causal reasoning. We follow constraint-based approaches to learning Bayes net structure, where learning is based on observed conditional dependencies and independencies between variables of interest (e.g., the data are of the form “X is dependent on Y given any assignment to variables S” or of the form “X is independent of Y given any assignment to variables S”). Applying learning criteria in this model leads to the following results. (1) The mind change complexity of identifying a Bayes net graph over variables V from either dependency data or from independency data are , the maximum number of edges. (2) There is a unique fastest mind-change optimal Bayes net learner for either data type; convergence speed is evaluated using Gold’s dominance notion of “uniformly faster convergence”. For dependency data, the optimal learner conjectures a graph if it is the unique Bayes net pattern that satisfies the observed dependencies with a minimum number of edges, and outputs “no guess” otherwise. For independency data, the optimal learner conjectures a graph if it is the unique Bayes net pattern that satisfies the observed dependencies with a maximum number of edges, and outputs “no guess” otherwise. We investigate the complexity of computing the output of the fastest mind-change optimal learner for either data type, and show that each of these two problems is NP-hard (assuming P = RP). To our knowledge these are the first NP-hardness results concerning the existence of a uniquely optimal Bayes net structure.