An important part of the LHC legacy will be precise limits on indirect effects of new physics, parameterized for instance in an Effective Field Theory (EFT). These measurements involve many parameters and observables and are often challenging for established analysis methods. We explain the structure of this “likelihood-free” inference problem, review a wooden “simulation" from the 19th century, and discuss why physicists usually don’t acknowledge these aspects (but perhaps should). We then present powerful new analysis techniques based on machine learning. In an example analysis of WBF Higgs production we show that they let us put stronger constraints on EFT parameters than established methods, demonstrating their potential to improve the new physics reach of the LHC legacy measurements. While developed for particle physics, these ideas can be applied to a broad class of problems in fields as diverse as cosmology, genetics, and epidemiology.