Many natural, engineered, and social systems can be represented using the framework of a layered network, where each layer captures a different type of interaction between the same set of nodes. The study of such multiplex networks is a vibrant area of research. Yet, understanding how to quantify the correlations present between pairs of layers, and more so present in their co-evolution, is lacking. Such methods would enable us to address fundamental questions involving issues such as function, redundancy and potential disruptions. Here we show first how the edge-set of a multiplex network can be used to construct an estimator of a joint probability distribution describing edge existence over all layers. We then adapt an information-theoretic measure of general correlation called the conditional mutual information, which uses the estimated joint probability distribution, to quantify the pairwise correlations present between layers. The pairwise comparisons can also be temporal, allowing us to identify if knowledge of a certain layer can provide additional information about the evolution of another layer. We analyze datasets from three distinct domains---economic, political, and airline networks---to demonstrate how pairwise correlation in structure and dynamical evolution between layers can be identified and show that anomalies can serve as potential indicators of major events such as shocks.