We examine models for unsupervised learning with concave log-likelihood functions. We begin with the most well-known example, IBM Model 1 for word alignment (Brown et al., 1993), and study its properties, discussing why other models for unsupervised learning are so seldom concave. We then present concave models for dependency grammar induction and validate them experimentally. Despite their simplicity, we find that initializing the dependency model with valence using our concave models can approach state of the art grammar induction results for English and Chinese. 1