  Using the mathematical framework of information geometry, we introduce a
novel method which allows one to efficiently determine the exact shape of
simultaneous confidence regions for non-linearly parametrised models.
Furthermore, we show how pointwise confidence bands around the model
predictions can be constructed from detailed knowledge of the exact confidence
region with little additional computational effort. We exemplify our methods
using inference problems in cosmology and epidemic modelling. An open source
implementation of the developed schemes is publicly available via the
InformationGeometry.jl package for the Julia programming language.
