AgentCBO: Causal Belief Routing for Bayesian Optimization under Unknown Graphs

Causal Bayesian optimization (CBO) typically relies on the unrealistic assumption of a perfectly known causal graph. Recent methods relax this constraint by coupling optimization with structure learning, but they often waste sample budgets on structural uncertainties that are irrelevant to the optimizer. In this paper, we formalize \emph{routing-equivalence} to shift the objective from full causal-graph recovery to causal belief routing, where structural differences only matter when they alter optimizer-facing objects such as admissible exploration sets and \emph{do}-priors. Furthermore, we propose \agentcbo{}, a graph-belief routing framework that incorporates LLM causal priors through guarded proposal and commit mechanisms. Specifically, a \textsc{Language-Prior Agent} combines LLM beliefs with constraint-based discovery to extract a protected causal backbone alongside residual uncertain edges. A \textsc{Calibration Agent} and a \textsc{Residual Repair Agent} then refine only these uncertain structures through score-gated bounded local edits. Finally, a deterministic commit gate evaluates candidate updates using routing-proxy checks before allowing them to influence intervention selection. Across ten benchmarks, \agentcbo{} performs comparably to known-graph CBO, demonstrating that guarded LLM priors can improve fixed-budget CBO performance.