Local Riemannian geometry of model manifolds and its implications for practical parameter identifiability

PLoS One.                 online article

When non-linear models are fitted to experimental data, parameter estimates can be poorly con- strained albeit being identifiable in principle. This means that along certain paths in parameter space, the log-likelihood does not exceed a given statistical threshold but remains bounded. This situation, denoted as practical non-identifiability, can be detected by Monte Carlo sampling or by systematic scanning using the profile likelihood method. In contrast, any method based on a Taylor expansion of the log-likelihood around the optimum, e.g., parameter uncertainty estimation by the Fisher Information Matrix, reveals no information about the boundedness at all. Here, based on two exemplary dynamic models we show that the information about the boundeds can already be con- tained in the Christoffel Symbols, which are computed from model sensitivities up to second order at the optimum. Assuming constant Christoffel Symbols in the geodesic equation, approximate Rie- mannian Normal Coordinates are constructed. The new coordinates give rise to an approximative log-likelihood, featuring flat directions and boundeds similar to that of the original log-likelihood. The approximative log-likelihood can be employed as a computationally efficient surrogate func- tion to replace the computationally demanding original objective function in parameter estimation problems and uncertainty analysis