Measuring the properties of the Higgs boson is one of the most important missions for Run 2 of the LHC. These measurements can be understood and optimized with intuitive and powerful tools from information geometry. The central object is the Fisher information matrix. It allows us to define the maximum precision with which theory parameters can be measured in a given experiment. We calculate the information on dimension-six operators in different Higgs signatures. We demonstrate how information geometry lets us improve event selections, determine the most powerful observables, and compare the power of multivariate techniques to traditional histogram-based analyses. In particular, we systematically analyze which LHC signatures offer dedicated CP measurements in the Higgs-gauge sector, and discuss the nature of the information they provide.