Hierarchical A∗ (HA∗ ) uses of a hierarchy of coarse grammars to speed up parsing without sacrificing optimality. HA∗ prioritizes search in refined grammars using Viterbi outside costs computed in coarser grammars. We present Bridge Hierarchical A∗ (BHA∗ ), a modified Hierarchial A∗ algorithm which computes a novel outside cost called a bridge outside cost.