Dynamical models based on relativistic fluid dynamics provide a powerful tool to extract the properties of the strongly-coupled quark-gluon plasma (QGP) produced by ultrarelativistic nuclear collisions. The largest source of uncertainty in these model-to-data extractions is the choice of theoretical initial conditions (ICs) used to model the distribution of energy or entropy at the hydrodynamic starting time. Descriptions of the ICs are generally improved through iterative cycles of testing and refinement. Individual models are compared to experimental data; the worst models are discarded and best models retained. Consequently, successful traits (assumptions) are passed on to subsequent generations of the theoretical landscape. This bottom-up approach correspondingly describes a form of theoretical trial and error, where each trial proposes an ab initio solution to the problem at hand. A natural complement to this strategy, is to employ a top-down or data-driven approach which is able to reverse engineer properties of the ICs from the constraints imposed by the experimental data. In this dissertation, I motivate and develop a parametric IC model based on a family of functions known as the generalized means. The ansatz closely mimics the variability of ab initio calculations and serves as a reasonable parametric form for exploring QGP energy and entropy deposition assuming imperfect knowledge of the complex physical processes which lead to its creation. With the parametric model in hand, I explore broad implications of the proposed ansatz using recently adapted Bayesian methods to simultaneously constrain properties of the ICs and QGP medium using experimental data from the Large Hadron Collider. These analyses show that the ICs are highly constrained by available measurements and provide evidence of a unified hydrodynamic description of small and large nuclear collision systems.