Mean-field interacting diffusion processes are ubiquitous: originally introduced as models of gases in statistical physics, they have found application in biology, economics, and machine learning, where instead of interacting particles they respectively describe animals, agents, or parameters. Daniel Lacker recently gave optimal rates for propagation of chaos—the expected property that finite sub-collections of particles become independent in the large particle limit—for interacting diffusions with a large class of interaction kernels. This work leverages the BBGKY hierarchy—a PDE hierarchy describing the marginal distributions—to give iterated bounds on the relative entropy between the marginals and the McKean-Vlasov equation. Motivated by this I will give higher-order corrections to propagation of chaos for systems with bounded interactions. The argument reframes propagation of chaos as a stability property of the BBGKY hierarchy and uses an L^2-type norm and cluster expansions. The main result also implies many interesting corollaries about the statistics of particles. Based on joint work with Keefer Rowan.