Effective Hamiltonians in tandem with statistical mechanics offer a rigorous connection between 0 Kelvin ab-initio simulations and finite temperature experimental observations. Specifically, cluster expansion Hamiltonians can extrapolate the complex, many-body electron problem of density functional theory (DFT) into a series of products-of-sites on an atomic scale. The resulting energy polynomial is computationally inexpensive, and hence suitable for the (tens of) thousands of calculations of thousand-site systems required by stochastic methods. We present a new method to run a cluster expansion "in reverse", taking high-temperature observations and using thermodynamic connections to predict the 0 Kelvin energy spectrum and associated ground states. By re-examining the cluster expansion formalism through the lens of entropy-maximization approaches, we develop an algorithm to select clusters and determine cluster interactions using only a few, high-temperature experiments on disordered phases. We will demonstrate our new approach, and also describe two systems we have studied using traditional DFT-backed cluster expansions: the Co-Pt binary alloy, and the MnRu2Sn-FeRu2Sn pseudo-binary.