Extending Multi-Source Bayesian Optimization With Causality Principles

Causal Bayesian Optimization (CBO) combines causal inference with Bayesian optimization to enable sample-efficient intervention selection in systems with causal structure. This survey provides a systematic review of CBO through a unified BO-loop perspective, showing how causal assumptions shape intervention search spaces, surrogate models, acquisition functions, and decision policies. We organize existing methods by graph and system-knowledge assumptions, environment, intervention representation, surrogate architecture, and decision rule, and connect CBO to causal bandits, Bayesian experimental design, safe optimization, policy search, and causal abstraction. We also introduce a reproducibility-oriented benchmark spanning hard- and soft-intervention settings, with standardized GAP and a new trajectory-aware Path-Aware GAP (PA-GAP), evaluating seven CBO methods and a non-causal BO baseline across thirteen datasets, three budgets, and two metrics. Results show that no method dominates uniformly: rankings depend on dataset, budget, metric, and how causal information is used, while strong non-causal baselines remain competitive in several settings. Controlled graph-misspecification and omitted-variable stress tests further show that rankings can change substantially when learner-side causal information is perturbed. We conclude by identifying key open challenges, including robustness to causal-assumption violations, scalable unknown-graph optimization, mixed intervention types, realistic cost models, stronger theoretical guarantees, and integration with modern representation learning and causal abstractions.

Bayesian optimization is a popular way to optimize expensive systems, where every experiment, simulation, or intervention costs time or money. In its standard form, it treats the variables we control as plain inputs to a black box and cannot tell apart mere correlation from a real cause and effect. Causal Bayesian optimization closes part of this gap by using a known causal graph together with observational data to decide which variables are worth intervening on. Existing methods, however, learn the effect of each possible intervention almost in isolation, even though in a causal system these effects usually share the same underlying mechanisms. We propose graph-coupled causal Bayesian optimization, which ties the different intervention effects together through the uncertainty we have about a small set of shared causal parameters. The result is a causal kernel that lets evidence collected from one intervention improve our estimate of related interventions. For identifiable linear Gaussian causal models, we show that this kernel has low rank, bounded by the number of shared parameters rather than by the size of the intervention menu. This in turn yields an information-gain bound that grows only logarithmically in the optimization horizon, and a regret bound that cleanly separates three sources of error: optimization, causal estimation, and the choice of which intervention sets to consider. We also describe nonlinear and adaptive extensions. Across theory-aligned Gaussian systems, shared-mechanism stress tests, and standard causal optimization benchmarks, the method keeps the benefits of causal Bayesian optimization while transferring information across related interventions, with the clearest gains when direct interventions on the target's parents are unavailable and sparse interventional data must be reused across a large family of candidate interventions.

Bayesian Optimisation (BO) under unknown constraints is particularly challenging when feasible regions are small. In such settings, existing methods that typically rely solely on evaluations of the true objective and constraints struggle to efficiently explore the design space. However, many real-world applications offer auxiliary data sources (e.g. surrogate models or simplified simulations) that can support early exploration. Despite this potential, their integration into constrained BO remains largely unexplored. We propose a general multi-source framework that extends constrained Max-value Entropy Search, capturing inter-source correlation while balancing evaluation cost and information gain. Experiments on both synthetic and physics-based benchmarks show that our method efficiently identifies feasible and optimal solutions, even when auxiliary data are only weakly correlated. The proposed approach consistently outperforms existing methods, particularly in early-stage exploration.