Bayesian inference of covariate-parameter relationships for population modelling

We consider population modelling using parametrised ordinary differential equation initial value problems (ODE-IVPs), in the setting where the parameter vector for each individual is not known, but a vector of covariates is available for each individual and follows an unknown distribution. For each individual, only one component of the solution to the ODE-IVP can be observed at only finitely many observation times. An important problem is to identify a covariate-parameter relationship that maps covariate vectors to parameter vectors. Such settings and problems arise in pharmacology, where the observations are blood drug concentrations, the covariates are clinically observable quantities, and the covariate-parameter relationship is used for personalised drug dosing. We use the framework presented in (Nickl, 2023) to analyse a nonparametric, nonlinear Bayesian inverse problem for the unknown covariate-parameter relationship, for a fixed finite time design, random covariate design, and a family of rescaled Gaussian priors. We provide sufficient conditions for posterior contraction, local asymptotic normality, and a Bernstein-von Mises theorem for linear functionals, and show theoretically that these conditions are met on a two-compartment model from the pharmacokinetics literature.