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Maximin Robust Bayesian Experimental Design

Hany Abdulsamad, Sahel Iqbal, Christian A. Naesseth, Takuo Matsubara, Adrien Corenflos

Abstract

We address the brittleness of Bayesian experimental design under model misspecification by formulating the problem as a max--min game between the experimenter and an adversarial nature subject to information-theoretic constraints. We demonstrate that this approach yields a robust objective governed by Sibson's $α$-mutual information~(MI), which identifies the $α$-tilted posterior as the robust belief update and establishes the Rényi divergence as the appropriate measure of conditional information gain. To mitigate the bias and variance of nested Monte Carlo estimators needed to estimate Sibson's $α$-MI, we adopt a PAC-Bayes framework to search over stochastic design policies, yielding rigorous high-probability lower bounds on the robust expected information gain that explicitly control finite-sample error.

Maximin Robust Bayesian Experimental Design

Abstract

We address the brittleness of Bayesian experimental design under model misspecification by formulating the problem as a max--min game between the experimenter and an adversarial nature subject to information-theoretic constraints. We demonstrate that this approach yields a robust objective governed by Sibson's -mutual information~(MI), which identifies the -tilted posterior as the robust belief update and establishes the Rényi divergence as the appropriate measure of conditional information gain. To mitigate the bias and variance of nested Monte Carlo estimators needed to estimate Sibson's -MI, we adopt a PAC-Bayes framework to search over stochastic design policies, yielding rigorous high-probability lower bounds on the robust expected information gain that explicitly control finite-sample error.
Paper Structure (30 sections, 19 theorems, 145 equations, 5 figures, 3 tables)

This paper contains 30 sections, 19 theorems, 145 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Expected Information Gain
  4. Variational Mutual Information
  5. Stochastic Design Policies
  6. Robust Experimental Design
  7. Maximin Experimental Design
  8. Robust Expected Information Gain
  9. Risk-Sensitive Interpretation
  10. Nested Monte Carlo Estimator
  11. PAC-Bayesian Design Policies
  12. Related Work
  13. Numerical Evaluation
  14. Discussion
  15. Organization of the Appendix
  16. ...and 15 more sections

Key Result

Lemma 1

Shannon's mutual information between $\theta$ and $x$ given $\xi$ is the minimal value of the Kullback--Leibler divergence between the true joint distribution and the product of variational marginals: The infimum is attained when the variational marginals coincide with the true marginals, so that $\mu^{\star}(\theta) = p(\theta)$ and $\nu^{\star}(x \mid \xi) = p(x \mid \xi)$. At this optimum, we

Figures (5)

  • Figure 1: Comparison of realized information gains under nominal and robust formulations for linear regression (left) and A/B testing (right). Histograms show the empirical distributions of gains obtained from $10^4$ simulations using optimal designs. The dashed line indicates the Sibson $\alpha$-mutual information benchmark.
  • Figure 2: Comparison of expected and actual coverage for nominal and robust posteriors given optimal and random designs in linear regression (left) and A/B testing (right). Nominal posteriors are overconfident, while robust posteriors are systematically conservative. Optimizing the design amplifies conservativeness further.
  • Figure 3: Empirical distributions of regret (left) and design optimality (right) across 1024 simulations for a naive optimizer and a PAC-Bayes policy for linear regression (top) an A/B testing (bottom). The naive optimizers exhibit higher regret and variability. Their designs are suboptimal, reflected in smaller design ratios relative to the theoretically optimal design.
  • Figure 4: Robust expected information gain for a two-dimensional linear regression problem with a correlated prior, as a function of $\alpha \in (0, 1)$. Contour lines depict the objective landscape over $(\xi_{1}, \xi_{2})$, highlighting how the optimal designs ($\star$) shift with $\alpha$.
  • Figure 5: Robust expected information gain as a function of $\alpha \in (0, 1)$ for an A/B testing problem with 25 participants. The optimal allocation, highlighted in dark gray, shifts as $\alpha$ varies.

Theorems & Definitions (25)

  • Definition 1: Expected information gain
  • Lemma 1: Shannon's variational mutual information, verdu2015alpha
  • Definition 2: Ambiguity set
  • Lemma 2: Minimization decomposition
  • Proposition 1: Robust expected information gain
  • Corollary 1: Worst-case generative process
  • Proposition 2: Uninformative design
  • Proposition 3: Robust conditional information gain
  • Corollary 2: Nested representation of $I_{\alpha}^{S}$
  • Definition 3: Nested Monte Carlo estimator $\tilde{I}_{\alpha}^{S}$
  • ...and 15 more