Index: Bayesian Experimental Design

Bayesian Experimental Design

Routing Summary

Information-theoretic design of experiments: choose designs $\xi$ to maximize the expected information gain (EIG) about latents $\theta$. This topic ingests four papers tracing the field’s full arc — its foundation (Lindley 1956), fast EIG estimation (Foster 2019), unified gradient design optimization (Foster 2020), and a review through policy-based adaptive design (Rainforth 2023). Contains 21 notes across 4 sub-topics.

• The big picture across all papers / where to start? → Bayesian Experimental Design - Overview
• Core concepts (EIG, nested estimation, adaptive design)? → Foundations
• Fast EIG estimators (posterior, marginal, VNMC, implicit)? → Variational EIG Estimators
• One-stage gradient design (ACE, PCE, high-D applications)? → Gradient-Based Unified BOED
• The state of the field (objectives, computation, policies, challenges)? → Modern BED Review

Sub-topics

• Foundations — COVERS: Lindley’s (1956) founding average-information measure (Defs 1–2, Theorems 1–9, the design rule); the EIG objective & its four equivalent (mutual-information) forms; double intractability and the NMC estimator ($\mathcal{O}(C^{-1/3})$); sequential/adaptive design and the incremental/total EIG. (4 notes.)
• Variational EIG Estimators — COVERS (Foster 2019): four amortized variational EIG estimators — posterior/Barber–Agakov (lower), marginal (upper), VNMC (upper, consistent), implicit-likelihood — with $\mathcal{O}(T^{-1/2})$ convergence and selection rules. (6 notes.)
• Gradient-Based Unified BOED — COVERS (Foster 2020): single SGA loop jointly optimizing a variational lower bound and the design; the ACE & PCE contrastive bounds; likelihood-free ACE and gradient estimators; high-dimensional applications (400-D regression, 100-D docking). (5 notes.)
• Modern BED Review — COVERS (Rainforth 2023): EIG vs Fisher-information objectives; the computational revolution (MLMC debiasing, variational, implicit); stochastic-gradient design; deep adaptive design (policies); open challenges. (6 notes.)

Cross-Cutting Concepts

Concepts that span multiple sub-topics:

• Expected Information Gain — the single objective every method optimizes: Expected Information Gain (defined) → estimated by Foster 2019 estimators / MLMC & variational → optimized by Foster 2020 SGA / DAD policies.
• Mutual-information bounds repurposed for design — Barber–Agakov (Variational Posterior Estimator (Barber-Agakov)), InfoNCE (Prior Contrastive Estimation (PCE)), and the adaptive contrastive bound (Adaptive Contrastive Estimation (ACE)).
• Bound-trapping — pairing a lower bound (ACE/${\hat{\mu }}_{\text{post}}$) with an upper bound (VNMC/${\hat{\mu }}_{\text{marg}}$) to verify designs: Convergence Rates and Estimator Selection, High-Dimensional Design Applications.
• Sequential / adaptive design — from greedy BAD (Sequential and Adaptive BED) to amortized non-myopic policies (From Designs to Policies (Deep Adaptive Design)).
• Implicit-likelihood models — Implicit Likelihood Estimator (error-bounded) → Likelihood-Free ACE and Gradient Estimation (bound-preserving) → review §3.3.2.

Concept Dependency Chain

Lindley’s Information Measure → Expected Information Gain → Nested Estimation and Nested Monte Carlo → Variational BOED - Overview → {Variational Posterior Estimator (Barber-Agakov), Variational Marginal Estimator, Variational NMC Estimator, Implicit Likelihood Estimator} → Convergence Rates and Estimator Selection → Unified SGD BOED - Overview → Adaptive Contrastive Estimation (ACE) → Prior Contrastive Estimation (PCE) / Likelihood-Free ACE and Gradient Estimation → High-Dimensional Design Applications; and (review thread) Information-Theoretic Design Objectives → The Computational Revolution in EIG Estimation → Optimization and Gradient Schemes for BED → From Designs to Policies (Deep Adaptive Design) → Open Challenges and Future Directions.

Sources

• Lindley 1956 - On a Measure of the Information Provided by an Experiment.pdf — Lindley, D.V., On a Measure of the Information Provided by an Experiment, Ann. Math. Stat. 27(4):986–1005, 1956. The founding paper.
• Foster et al 2019 - Variational Bayesian Optimal Experimental Design.pdf — Foster et al., Variational Bayesian Optimal Experimental Design, NeurIPS 2019. arXiv:1903.05480.
• Foster et al 2020 - Unified Stochastic Gradient BOED.pdf — Foster et al., A Unified Stochastic Gradient Approach to Designing Bayesian-Optimal Experiments, AISTATS 2020. arXiv:1911.00294.
• Rainforth et al 2023 - Modern Bayesian Experimental Design.pdf — Rainforth, Foster, Ivanova, Bickford Smith, Modern Bayesian Experimental Design, Statistical Science 2023. arXiv:2302.14545.

See Also

• Bayesian Statistics — inference, computation, and decision analysis that BED builds on
• Experimental Design (frequentist) — the classical power-analysis counterpart
• Decision Analysis — EIG as expected utility of an experiment