Link prediction, or the inference of future or missing connections between entities, is a well-studied problem in network analysis. A multitude of heuristics exist for link prediction in ordinary networks with a single type of connection. However, link prediction in multiplex networks, or networks with multiple types of connections, is not a well understood problem. We propose a novel general framework and three families of heuristics for multiplex network link prediction that are simple, interpretable, and take advantage of the rich connection type correlation structure that exists in many real world networks. We further derive a theoretical threshold for determining when to use a different connection type based on the number of links that overlap with an Erdős-Rényi random graph. Through experiments with simulated and real world scientific collaboration, transportation and global trade networks, we demonstrate that the proposed heuristics show increased performance with the richness of connection type correlation structure and significantly outperform their baseline heuristics for ordinary networks with a single connection type.