Abstract
One of the main goals of mathematical modelling in systems biology 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 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.
In the non-parametric case, the maximum likelihood estimate is known to overfit the data, an issue that is commonly tackled by regularization. However, the majority of regularizations are ad-hoc choices which lack invariance under re-parametrization of the model and hence result in inconsistent estimates for equivalent models.
We introduce the empirical reference prior, a non-parametric, transformation-invariant estimator for the prior distribution, which represents a symbiosis between the objective and empirical Bayes methodologies.
We demonstrate the performance of this approach by applying it to an ordinary differential equation model for the human menstrual cycle, a typical example from systems biology modelling. In particular, we show how this method supports the construction of a virtual patient population for model-based treatment computation in reproductive endocrinology.