A large part of physics can be thought of as an attempt to infer the nature of a system's state from imprecise coarse-grained measurements. This is an ill-posed problem because certain features of the state cannot be resolved by the measurements. However, a model can nonetheless be made of those features which can be observed. We explain how those relevant parameters can be concretely characterized using quantum distinguishability metrics, and show that Gaussian states emerge as relevant manifolds of effective states. This framework allows us to explain in elementary terms some of the problems occurring in quantum field theory, and their solutions.