The geometry of Sloppiness

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.

Causal Geometry

“…This allows giving a continuum formulation of causal emergence, as well as a novel local measure of causal optimality. On the other side, our work establishes the proper formal role of interventions and causality in sloppiness, potentially resolving a long-standing formal challenge around the non-covariance of metric eigenvalues [ 28 ] and showing that not only the hyper-ribbon manifold structure, but also its relation to intervention capabilities account for the emergence of simple models.…”

“…On the flip side, our construct provides a novel formulation of information geometry that allows it to explicitly account for the causal relations of a model. Moreover, we argue that this is necessary for formal consistency when working with the Fisher information matrix eigenvalues (namely, their covariant formulation; see Section 3.2 ) [ 28 ]. This suggests that model reduction based on these eigenvalues may not be made fully rigorous without explicitly accounting for causal relations in the model.…”

“…The trouble with this approach is that the components of the matrix , and hence its eigenvalues, depend on the particular choice of -coordinates on the effect manifold [ 6 , 28 ]. Since the parameters represent some conceptual abstraction of the physical system, they constitute an arbitrary choice.…”

“…Here, we show that by explicitly considering intervention capabilities, we can construct a local, but still coordinate independent sloppiness metric. This becomes possible since interventions give a second independent distance metric on [ 28 ]. The matrix product appearing in Equation ( 8 ) is then a linear transformation, and thus, its eigenvalues are coordinate independent.…”

Identifiability analysis for stochastic differential equation models in systems biology

“…Much of the usefulness of calibrated models hinges on an assumption that model parameters are identifiable. Heavily over-parameterised models, with large numbers of practically non-identifiable parameters, are often referred to as sloppy in the systems biology literature [138][139][140][141]. Worryingly, these issues of parameter identifiability can often go undetected: models with non-identifiable parameters can still match experimental data (figure 8), but may have poor predictive power and provide little or no mechanistic insight [28].…”