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Covariance constraints for stochastic inverse problems of computer models

Nicolas Bousquet, Mélanie Blazère, Thomas Cerbelaud

TL;DR

New prior constraints are derived using the principles of global sensitivity analysis and information theory that reflect the idea that the solution should explain most of the observable uncertainty, while the model noise remains a secondary factor of this uncertainty.

Abstract

Stochastic inverse problems considered in this article consist of estimating the probability distributions of intrinsically random inputs of computer models. These estimations are based on observable outputs affected by model noise, and such problems are increasingly examined in parametric Bayesian contexts where the parameters of the targeted input distributions are affected by epistemic uncertainties. With the aim of improving the meaningfulness of solutions found by statistical algorithms -- in the sense that forward simulations based on such solutions must lead to relevant observables -- we derive new prior constraints using the principles of global sensitivity analysis and information theory. Primarily formalized as constraints on covariances in Gaussian linear or linearizable situations, they reflect the idea that the solution should explain most of the observable uncertainty, while the model noise remains a secondary factor of this uncertainty. Simulated experiments highlight that, when injected into stochastic inversion algorithms, these constraints can indeed limit the influence of model noise on the result. They provide hope for future extensions in more general frameworks, for example through the use of linear Gaussian mixtures.

Covariance constraints for stochastic inverse problems of computer models

TL;DR

New prior constraints are derived using the principles of global sensitivity analysis and information theory that reflect the idea that the solution should explain most of the observable uncertainty, while the model noise remains a secondary factor of this uncertainty.

Abstract

Stochastic inverse problems considered in this article consist of estimating the probability distributions of intrinsically random inputs of computer models. These estimations are based on observable outputs affected by model noise, and such problems are increasingly examined in parametric Bayesian contexts where the parameters of the targeted input distributions are affected by epistemic uncertainties. With the aim of improving the meaningfulness of solutions found by statistical algorithms -- in the sense that forward simulations based on such solutions must lead to relevant observables -- we derive new prior constraints using the principles of global sensitivity analysis and information theory. Primarily formalized as constraints on covariances in Gaussian linear or linearizable situations, they reflect the idea that the solution should explain most of the observable uncertainty, while the model noise remains a secondary factor of this uncertainty. Simulated experiments highlight that, when injected into stochastic inversion algorithms, these constraints can indeed limit the influence of model noise on the result. They provide hope for future extensions in more general frameworks, for example through the use of linear Gaussian mixtures.

Paper Structure

This paper contains 26 sections, 18 theorems, 138 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Formalism and main notations
  3. A preliminary: Two intuitive notions of meaningful solutions
  4. Linear models
  5. Locally linearizable models
  6. Fisher inversion constraints
  7. Formalisation
  8. Gaussian linear models
  9. Fisher matrix characterization
  10. Conditions for Fisher constraints
  11. The case $\Gamma=\tau^2 I_p$.
  12. Gaussian linearizable models
  13. Theoretical approximation
  14. Computational aspects
  15. Numerical experiments
  16. ...and 11 more sections

Key Result

Proposition 1

The stochastic inverse problem (inversion.problem-eq:model_toy) is meaningful in Sobol' sense if and only if It is meaningful in entropic sense if and only if which imposes that $a\Gamma a^T$ be invertible.

Figures (3)

  • Figure 1: Summary of posterior relative means and 90% credibility intervals with respect to simulation values, on average on the values of $d$, for $n=30$. The inversion algorithm considers Situation 1 ($q=1$). Means are perfect estimates of simulation values when the $y-$axis values reach 100%.
  • Figure 3: Boxplots of errors $e^{(k)}_{S_j}$ (KL divergences) between classes of models and the simulation model (Situation 1: $q=1$), for two different data sizes.
  • Figure 5: Examples of the joint posterior predictive distribution of $Y$ obtained under the considered settings. The black dots represents simulated observations $Y$. The left column uses the original model $g$ and the right column uses a linear surrogate.

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Corollary 1
  • Proposition 2
  • Proposition 3
  • Example 1
  • Definition 4
  • Theorem 1
  • ...and 28 more