Theoretical Statistics and Mathematics Unit, ISI Delhi
In several fields like genetics, viral dynamics, pharmacokinetics and pharmacodynamics, population studies and so on, regression models are often given by differential equations which are not analytically solvable. In this talk, Bayesian estimation and uncertainty quantification is addressed in such models. The approach is based on embedding the parametric model in 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 given 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 a simple direct sampling. We establish Bernstein-von Mises type theorems for the induced posterior distribution of the model parameters. We consider different choices of the projection map and study its impact on the asymptotic efficiency of the Bayesian estimator. We further show that posterior credible regions have asymptotically correct frequentist coverage. A simulation study and applications to some real date sets show practical usefulness of the method. Ideas of extending the methods to higher order differential equations and partial differential equations will also be discussed.
This talk is based on joint work with Prithwish Bhaumik.