We investigate 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 analyze 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. We find our concave models to be effective initializers for the dependency model of Klein and Manning (2004) and show that we can encode linguistic knowledge in them for improved performance.