Goodness-of-fit testing for nonlinear inverse problems with random observations

TL;DR

This paper develops nonparametric goodness-of-fit tests for nonlinear inverse problems with random data, demonstrating Bayesian posterior contraction, convergence, and distinguishability, especially applied to pharmacokinetics models.

Contribution

It introduces a Bayesian framework for goodness-of-fit testing in nonlinear inverse problems, establishing contraction rates and distinguishability results with practical applications.

Findings

Bayesian posterior contracts at a specific rate uniformly over true parameters.

Posterior mean converges uniformly satisfying a concentration inequality.

Distinguishability of bounded alternatives using posterior-based tests is proven.

Abstract

This work is concerned with nonparametric goodness-of-fit testing in the context of nonlinear inverse problems with random observations. Bayesian posterior distributions based upon a Gaussian process prior distribution are proven to contract at a certain rate uniformly over a set of true parameters. The corresponding posterior mean is shown to converge uniformly at the posterior contraction rate in the sense of satisfying a concentration inequality. Distinguishability for bounded alternatives separated from a composite null hypothesis at the posterior contraction rate is established using infimum plug-in tests based on the posterior mean and also on maximum a posteriori estimators. The results are applied to a class of inverse problems governed by ordinary differential equation initial value problems that is widely used in pharmacokinetics. For this class, uniform posterior contraction rates are proven and then used to establish distinguishability.