Ultra-relativistic heavy-ion collisions produce a hot and dense fireball of deconfined quarks and gluons known as a quark-gluon plasma (QGP). Since its discovery, the QGP has exhibited a number of interesting and unexpected properties. Most notably, it was found to behave like a strongly coupled liquid with astonishingly small specific shear viscosity, prompting some to call it the “perfect fluid.” Hydrodynamic computer models allow researchers to simulate the time-evolution of QGP droplets produced in relativistic heavy-ion collisions. The models generate events as they might occur inside the detector and output mock data that can be directly compared with experiment. Free parameters of the model, e.g. the medium's viscous transport coefficients, are then tuned to optimally replicate experiment and infer intrinsic properties of the QGP medium. A prominent source of uncertainty in determining QGP transport coefficients is in modeling the initial stages of the collision. Model predictions vary with the choice of initial conditions and hence prefer hydrodynamic transport coefficients, which differ from calculation to calculation. The ongoing effort to improve current estimates of QGP transport coefficients typically focuses on improving theoretical descriptions of the initial conditions and introducing new, sensitive observables to assess the validity of each model's inherent assumptions and approximations. In this talk I introduce an alternate approach to ab initio theoretical calculations based on Bayesian parameter estimation, which extracts the QGP initial conditions directly from experimental data. Starting from a minimal set of theoretical assumptions, we parameterize entropy deposition in ultra-relativistic nuclear collisions and show that the parametric initial conditions are over-constrained by simultaneous fits to multiple experimental observables. We use these constraints to refine our current understanding of the QGP initial conditions and provide refined credibility intervals on fundamental QGP parameters.