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

Remo Kretschmann, Han Cheng Lie

TL;DR

The paper tackles nonparametric goodness-of-fit testing for nonlinear inverse problems with random observations under a Bayesian framework with Gaussian process priors. It develops non-asymptotic, uniform results: (i) posterior contraction at rate $\delta_N$ and uniform convergence of the posterior mean in several normed spaces, and (ii) distinguishability at the contraction rate for infimum plug-in tests based on both posterior means and MAP estimators. The authors prove these results in $L^2_\zeta$, Sobolev, and supremum norms, and apply them to inverse problems governed by linear ODE-IVPs, notably a two-compartment pharmacokinetic model, obtaining explicit bounds on testing errors. The work also connects to and extends prior results on posterior contraction and testing in nonlinear and random-design settings, offering a comprehensive, quantitative framework for minimax-type separation analysis in nonlinear indirect problems with practical pharmacokinetic applications.

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.

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

TL;DR

The paper tackles nonparametric goodness-of-fit testing for nonlinear inverse problems with random observations under a Bayesian framework with Gaussian process priors. It develops non-asymptotic, uniform results: (i) posterior contraction at rate and uniform convergence of the posterior mean in several normed spaces, and (ii) distinguishability at the contraction rate for infimum plug-in tests based on both posterior means and MAP estimators. The authors prove these results in , Sobolev, and supremum norms, and apply them to inverse problems governed by linear ODE-IVPs, notably a two-compartment pharmacokinetic model, obtaining explicit bounds on testing errors. The work also connects to and extends prior results on posterior contraction and testing in nonlinear and random-design settings, offering a comprehensive, quantitative framework for minimax-type separation analysis in nonlinear indirect problems with practical pharmacokinetic applications.

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.
Paper Structure (31 sections, 29 theorems, 201 equations)

This paper contains 31 sections, 29 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Literature review
  3. Contributions
  4. Structure
  5. Notation
  6. Motivating example
  7. Inference for covariate-to-parameter mappings
  8. Goodness-of-fit testing
  9. Covariates and mechanistic model
  10. Distinguishability for plug-in tests
  11. Concentration inequalities in literature
  12. Nonlinear inverse problems with random observations
  13. Uniform convergence of posterior mean under posterior contraction
  14. Uniform posterior contraction and distinguishability in $L^2_\zeta$ norm
  15. Uniform posterior contraction and distinguishability in Sobolev and supremum norms
  16. ...and 16 more sections

Key Result

Lemma 3.4

Let $S\subseteq \Theta$ be nonempty, let $(t_N)_{N\in\mathbb{N}}$ be a sequence of nonnegative numbers, and let $(\widehat{\theta}_N)_{N\in\mathbb{N}}$ be a sequence of estimators. Then the type $1$ and type $2$ errors of the tests $(\Psi_N)_{N\in\mathbb{N}}$ from eq_infimum_plug_in_tests for testin satisfy

Theorems & Definitions (64)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3: Uniform stochastic boundedness
  • Lemma 3.4: Bound on type $1$ and type $2$ errors
  • proof : Proof of \ref{['errors_plug_in_general']}
  • Theorem 3.5: Distinguishability under uniform stochastic boundedness
  • proof : Proof of \ref{['theorem_distinguishability_under_uniform_stochastic_boundedness']}
  • Corollary 3.6: Distinguishability at slower rates
  • Remark 3.7
  • proof : Proof of \ref{['corollary_distinguishability_at_slower_rates']}
  • ...and 54 more