Empirical Bayes Methods for Prior Estimation in Systems Medicine

One of the main goals of mathematical modeling in systems medicine related to medical applications is to obtain patient-specific parameterizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Therefore, before applying Bayes' rule \emph{separately} to the data of each patient (which is typically performed using a non-informative prior), it is meaningful to use empirical Bayes methods in order to construct an informative prior from all available data. We compare the performance of four priors -- a non-informative prior and priors chosen by nonparametric maximum likelihood estimation (NPMLE), by maximum penalized likelihood estimation (MPLE) and by doubly-smoothed maximum likelihood estimation (DS-MLE) -- by applying them to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modeling.