Generalised Bayesian Optimal Experimental Design (GBOED)

• GBOED is a decision-theoretic framework that generalizes Bayesian design by replacing the traditional likelihood with flexible loss functions.
• It employs novel utility criteria and computational strategies like nested Monte Carlo and variational inference to manage model misspecification and high-dimensionality.
• The framework supports robust and goal-oriented experimental design, yielding improved estimation accuracy and practical benefits in diverse applications.

Generalised Bayesian Optimal Experimental Design (GBOED) is a decision-theoretic framework that extends classical Bayesian experimental design by relaxing key assumptions underlying the statistical model and inference procedure. In contrast to standard approaches that require a correctly specified likelihood and focus on expected information gain (EIG) for parameter recovery, GBOED accommodates misspecification, alternative loss functions, and wider experimental objectives—including robustness, geometric discrepancy, model discrimination, and goal-oriented utility. Research in this area has established rigorous information-theoretic foundations, derived surrogate optimality criteria, and introduced computational tools for tractable optimization in high and infinite-dimensional regimes, as well as for likelihood-free models and design adaptivity.

1. General Theory: Bayesian and Generalised Posterior Updates

In conventional Bayesian OED, the modeler posits a prior π(θ) and a parametric likelihood p(y|θ,d) connecting unknown parameters θ to potential experimental outcomes y under design d. The posterior is

$$\pi(\theta|y,d) \propto p(y|\theta,d)\ \pi(\theta),$$

and designs are selected to maximize a decision-theoretic utility, most commonly EIG:

$$\text{EIG}(d) = \mathbb{E}_{y\sim p(y|d)}\big[\mathrm{KL}(\pi(\cdot|y,d) \| \pi(\cdot))\big].$$

GBOED reframes this update by employing generalised Bayesian (Gibbs) posteriors (Barlas et al., 10 Nov 2025, Overstall et al., 2023). Here, the likelihood is replaced by an exponential weight derived from a loss ℓ(θ;y,d), and a learning-rate (temperature) parameter w>0:

$$\pi_w(\theta|y,d) = \frac{\exp(-w\,\ell(\theta;y,d))\,\pi(\theta)}{\int \exp(-w\,\ell(\theta;y,d))\,\pi(\theta)\,d\theta}.$$

When ℓ(θ;y,d) = -\log p(y|θ,d), w=1, the classical Bayesian update is recovered.

This generalisation allows the use of loss functions not directly linked to any likelihood, supporting robust alternatives (e.g., score matching, divergences such as β- or γ-divergence), and different calibration of learning rate to control the influence of data.

2. Generalised Utilities, Robustness, and Information Criteria

GBOED advances several utility criteria beyond standard EIG to capture experimental aims under model uncertainty and other goals:

Criterion	Mathematical Formulation	Usage/Interpretation
Gibbs EIG	$\mathrm{EIG}_w(d)$ as the pseudo-mutual information under the Gibbs posterior	Robust info gain, respects alternative loss
Robust/Model-Free Utility	$$U(d) = \mathbb{E}_{r\sim\mathcal{C}}\mathbb{E}_{y\sim \mathcal{D}(r;d)}[u(d,\theta^*_D(r;d),y)]$$	Aims at target fitted under "designer" distribution
Wasserstein EIG (W$_p$)	$$J_p(d) = \mathbb{E}_{y\sim p(y|d)}[W_p(\pi(\cdot|y,d),\pi(\cdot))]$$	Geometric, rather than entropic, spread
Goal-Oriented EIG	$$U(d) = \iint p(z,y|d)\log\frac{p(z|y,d)}{p(z)}dz\,dy$$	Info gain on functionals of θ (QoIs)
Model Discrimination	$$I(y;m|x) = \sum_{m} p(m)\int p(y|m,x)\log \frac{p(y|m,x)}{p(y|x)}dy$$	Mutual info between observation and model index

GBOED frameworks enable direct robustness to outliers and misspecification, with the learning rate or loss controlling the flattening of the posterior and lowering the sensitivity to data that the working model poorly explains. The selection of the appropriate loss function ℓ is central to the robustness guarantees, with bounded-influence functions achievable through proper scoring rules and weighted score-matching losses.

3. Computational Schemes and Optimisation Methodologies

The complexity of GBOED necessitates computational strategies adapted to the chosen utility and surrogate posterior. The main methodologies include:

• Nested Monte Carlo (NMC) for Gibbs EIG (Barlas et al., 10 Nov 2025), estimating inner integrals over both θ and y, with self-normalized importance sampling to ensure stability.
• Variational Inference for high-dimensional θ, optimising families q_ϕ(θ) to approximate the Gibbs posterior by minimizing KL divergence, and back-propagating gradients w.r.t. both variational and design parameters.
• Gaussian Process Emulation of the utility surface, especially in GLM contexts (Woods et al., 2016), to interpolate expensive Monte Carlo or sample-based utility evaluations.
• Coordinate Exchange (ACE) and global optimizers (e.g. Bayesian optimisation UCB/EI wrappers) to perform high-dimensional or blockwise search over design spaces (Woods et al., 2016, Overstall et al., 2023).
• Optimal Transport-based Approaches for Wasserstein criteria, leveraging closed-form solutions for linear-Gaussian models or numerically solving Monge–Ampère equations and their discretizations otherwise (Helin et al., 14 Apr 2025).
• Transport Map Surrogates (tensor trains/Knothe–Rosenblatt mappings) to generate conditional samples and densities with amortized cost, enabling fast evaluation of utility functions in sequential or high-dimensional settings (Koval et al., 2024).
• Stochastic Gradient Optimization entails formulating variational lower bounds (Barber–Agakov, ACE, PCE) as differentiable objectives and using (joint) stochastic gradient ascent in design and variational parameters, allowing scalable design search (Foster et al., 2019, Zaballa et al., 2023).
• Likelihood-Free/Ratio-Based Methods (ABC, density-ratio estimation, mutual-information neural lower bounds) for models where the likelihood is implicit or intractable, but one can generate (θ, y) samples via simulation (Chakraborty et al., 2024, Valentin et al., 2021, Zaballa et al., 2023).

Each approach presents trade-offs in stability, sample efficiency, memory usage, and suitability for particular design/topology regimes or model classes.

4. Theoretical Guarantees and Robustness Properties

General theoretical results across GBOED frameworks include:

• Non-negativity and Symmetry: Gibbs EIG and Wasserstein-based criteria are genuine distances or divergences between prior and posterior (or their pseudo-variants), ensuring the utility is non-negative and symmetric in its arguments (Barlas et al., 10 Nov 2025, Helin et al., 14 Apr 2025).
• Robustness to Misspecification: For properly chosen ℓ and w<1, GBOED mitigates bias and overconfidence under model misspecification, heavy tails, or outlier contamination, as predicted by limit theorems on the concentration of Gibbs posteriors to loss minimizers (Barlas et al., 10 Nov 2025).
• Stability Under Approximation: Bounds on the change in Wasserstein utility (e.g., in W$_1$ or W$_2$) under perturbations to the prior or likelihood demonstrate ε-stability of the criterion and resulting design (Helin et al., 14 Apr 2025).
• Error Analysis and Convergence Rates: Polynomial chaos surrogates and surrogate-based global optimisation exhibit sample complexity rates typical of regression or approximation theory, and sequential map-based methods allow amortized cost convergence in sequential design settings (Koval et al., 2024, Tarakanov et al., 2020).
• Effect of Covariate Shift: Under covariate shift (when the distribution of test designs differs from train designs), the generalization error of design-selected models can be decomposed into misspecification bias, estimation bias, and an error amplification term. Explicit incorporation of representativeness measures into the acquisition function improves out-of-distribution generalization performance (Tang et al., 9 Jun 2025).

5. Practical Guidance: Loss Selection, Calibration, and Implementation

Important implementation considerations and empirical findings include:

• Loss Function Choice: Negative log-likelihood is suitable for reasonably specified models; if misspecification or heavy-tailed data are suspected, employ robust losses such as weighted score matching with an adaptive IMQ kernel or proper divergence functions (Barlas et al., 10 Nov 2025).
• Learning Rate (w): Set w≈1 when the model is trusted; reduce w for more robustness, which induces exploration and avoids EIG over-exploitation. Cross-validation or pilot studies to select w can maintain divergence within a tolerable range (Barlas et al., 10 Nov 2025).
• Estimator Stability: Self-normalized importance sampling in nested MC prevents numerical instability; variational methods are imperative in high-dimensional parameter regimes (Barlas et al., 10 Nov 2025, Foster et al., 2019).
• Design Optimisation: Coordination between design selection and robust inference is crucial—running GBOED utility for design search and then classical inference does not recover the desired robustness, i.e., adapting both design and inference is critical (Barlas et al., 10 Nov 2025).
• Sequential/Adaptive Design: Map-based and transport approaches lend themselves naturally to sequential experiments, updating transport maps or posteriors at each stage for greedy or non-myopic design (Koval et al., 2024).
• Computational Scalability: Surrogate-based methods (GPs, polynomial chaos), differentiable variational models, and stochastic gradient ascent materially improve tractability in high-dimensional design or parameter spaces and avoid cubic scaling in classical surrogate-based BOED (Foster et al., 2019, Kennamer et al., 2022, Tarakanov et al., 2020).
• Goal-Orientation and Likelihood-Free Settings: Mutual information can be adapted to goal-oriented design (for QoIs that are functionals of θ), and likelihood-free inference is integrated via density-ratio estimation or ABC-driven estimators (Chakraborty et al., 2024, Zaballa et al., 2023).

6. Applications and Empirical Performance

GBOED frameworks have demonstrated substantial performance benefits:

• Linear Regression under Contamination: GBOED with robust losses offers 30–50% lower RMSE and NLL under 30% outlier contamination compared to classical BOED, maintaining exploration and avoiding design clustering (Barlas et al., 10 Nov 2025).
• Pharmacokinetic Models: In the presence of heavy-tailed additive and multiplicative noise, GBOED improves predictive accuracy and generalization metrics over BOED (Barlas et al., 10 Nov 2025).
• Subsurface Flow and Inverse Problems: PCE and transport-map methods maintain scalable design optimization for large-scale PDE models, achieving variance reduction and higher information-gain at modest computational cost (Tarakanov et al., 2020, Koval et al., 2024).
• Simulations and Non-Differentiable Forward Models: Particle-based, gradient-free, and amortized variational techniques demonstrate scalable design in complex models where no likelihood gradient information is available (Gruhlke et al., 17 Apr 2025, Zaballa et al., 2023).
• Adaptive Experimental Strategies: Sequential GBOED with surrogate updates in each round allows efficient exploitation/exploration trade-off and substantial posterior variance reduction across practical case studies (Koval et al., 2024, Barlas et al., 10 Nov 2025).
• Goal-oriented and Symbolic Discovery: GBOED enables model discrimination and downstream QoI-focused design via mutual information and entropy maximizing criteria, supporting symbolic regression and decision-awareness (Clarkson et al., 2022, Chakraborty et al., 2024).

7. Connections, Limitations, and Current Directions

Contemporary GBOED research synthesizes ideas from statistical decision theory, generalised Bayesian inference, optimal transport, variational inference, and robust statistics. The unifying principle is the replacement of hard likelihood-based design objectives with flexible, loss-driven, and robustness-oriented utilities, supported by scalable optimization algorithms.

Limitations in current methods include:

• Estimator instability and bias (especially for small w or imperfect variational/posterior approximations).
• Scaling challenges in very high-dimensional data or design spaces if surrogates or ratio estimators lose accuracy (Chakraborty et al., 2024, Foster et al., 2019).
• The amplification term in covariate shift generalization decomposition remains unobservable without access to the true DGP, so representativeness-based corrections are a principled but partial remedy (Tang et al., 9 Jun 2025).
• Need for sophisticated conditional generative models in transport/diffusion-based BOED beyond Gaussian or linear settings (Iollo et al., 2024).
• Open questions in sequential, non-myopic OED, adaptation to fully implicit models, and direct optimization of de-amplification in the presence of unquantified model error (Tang et al., 9 Jun 2025, Iollo et al., 2024).

Active areas of study include amortized and policy-based design, new MI surrogate criteria (Wasserstein EIG, contrastive MI bounds), further generalization theory, and expansion to practical high-dimensional scientific models and real-world experimental systems.