  One major challenge for the legacy measurements at the LHC is that the
likelihood function is not tractable when the collected data is
high-dimensional and the detector response has to be modeled. We review how
different analysis strategies solve this issue, including the traditional
histogram approach used in most particle physics analyses, the Matrix Element
Method, Optimal Observables, and modern techniques based on neural density
estimation. We then discuss powerful new inference methods that use a
combination of matrix element information and machine learning to accurately
estimate the likelihood function. The MadMiner package automates all necessary
data-processing steps. In first studies we find that these new techniques have
the potential to substantially improve the sensitivity of the LHC legacy
measurements.
