The geometry of sloppiness

TL;DR

This paper establishes a rigorous mathematical framework for the concept of sloppiness in models, relating it to model manifolds and structural identifiability, and proposes alternative ways to quantify sloppiness beyond traditional methods.

Contribution

It provides a precise mathematical foundation for sloppiness, redefining it as a comparison between measurement-induced premetric and a reference metric, expanding its quantification methods.

Findings

Redefines sloppiness in terms of premetric and reference metric

Links sloppiness to structural identifiability concepts

Applies framework to various parametric models

Abstract

The use of mathematical models in the sciences often involves the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. In this paper, we develop a precise mathematical foundation for sloppiness and define rigorously its key concepts, such as `model manifold', in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric. This opens up the possibility of alternative quantification of sloppiness, beyond the standard use of the Fisher Information Matrix, which assumes that parameter space is equipped with the usual Euclidean metric and the measurement error is infinitesimal. Applications include parametric statistical models, explicit time dependent models, and ordinary differential equation models.