Bayesian Smooth-and-Match strategy for ordinary differential equations models that are linear in the parameters

In many fields of application, dynamic processes that evolve through time are well described by systems of ordinary differential equations (ODEs). The analytical solution of the ODEs is often not available and different methods have been proposed to infer these quantities: from numerical optimization to regularized (penalized) models, these procedures aim to estimate indirectly the parameters without solving the system. We focus on the class of techniques that use smoothing to avoid direct integration and, in particular, on a Bayesian Smooth-and-Match strategy that allows to obtain the ODEs' solution while performing inference on models that are linear in the parameters. We incorporate in the strategy two main sources of uncertainty: the noise level in the measurements and the model error. We assess the performance of the proposed approach in three different simulation studies and we compare the results on a dataset on neuron electrical activity.