We first show how a structural locality bias can improve the accuracy of state-of-the-art dependency grammar induction models trained by EM from unannotated examples (Klein and Manning, 2004). Next, by annealing the free parameter that controls this bias, we achieve further improvements. We then describe an alternative kind of structural bias, toward “broken” hypotheses consisting of partial structures over segmented sentences, and show a similar pattern of improvement. We relate this approach to contrastive estimation (Smith and Eisner, 2005a), apply the latter to grammar induction in six languages, and show that our new approach improves accuracy by 1–17% (absolute).