The geometry of sloppiness

Mathematical models in the sciences often require the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. We develop the precise mathematical foundation for sloppiness and define rigorously its key concepts, such as `model manifold' in relation to concept of structural identifiability. The traditional definition of sloppiness uses the Fisher Information Matrix, and as such it deals with infinitesimal measurement error. We generalize sloppiness and define it in terms of the premetric on parameter space induced by measurement noise. Applications include parametric statistical models, explicit time dependent models, and ordinary differential equation models with time series data.