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In several fields like genetics, viral dynamics, pharmacokinetics and pharmacodynamics, population studies, and so on, regression models are often given by differential equations that are not analytically solvable. In this talk, Bayesian estimation and uncertainty quantification are addressed in such models. The approach is based on embedding the parametric nonlinear regression model into a nonparametric regression model and extending the definition of the parameter beyond the original model. The nonparametric regression function is expanded in a basis and normal priors are put on coefficients leading to a normal posterior, which then induces a posterior distribution on the model parameters through a projection map. The posterior can be obtained by simple direct sampling. We show that the posterior distribution of the model parameters is approximately normal. The most important consequence of this result is that the frequentist coverage of Bayesian credible regions approximately matches with their credibility levels, implying that the Bayesian and the frequentist measures of uncertainty quantification approximately agree. We consider different choices of the projection map and study their impact on the asymptotic efficiency of the Bayesian estimator. A simulation study and applications to some real data sets show the practical usefulness of the method. Extensions of the results to generalized regression and to higher order differential equation and partial differential equation models will also be discussed. |