When and Why LLM Causal Priors Help: Closed-Loop Prior Selection for Amortized Causal Inference
Causal effect estimation asks how an outcome would change under an intervention, and medicine, economics, and public policy all treat it as a foundational task. Prior-data fitted networks (PFNs) amortize the task: a model trained on large numbers of programmatically generated synthetic causal tasks reads a new problem's observational data into context and returns an interventional-effect estimate in a single forward pass. The capability of such models is largely determined by the synthetic training prior, which is currently designed by hand, a bottleneck acknowledged by both Do-PFN and CausalPFN. Large language models (LLMs) can now ``draw'' plausible causal graphs for a given domain, suggesting that LLM-distilled graphs could serve as prior material. Whether injecting such graphs helps at all, where any gain comes from, and when injection helps. Practice has so far relied on manual trial and error. We propose a \emph{closed-loop prior selection framework} that casts prior injection as a budget-constrained optimization over a candidate prior pool. Candidates undergo cheap post-training and are scored by a composite metric dominated by real-domain generalization; the winner then receives full training and paired statistical validation. On a 7.34M-parameter Do-PFN, the framework's winner attains a formally significant $2.75\times$ gain on the primary evaluation domain, and its error falls below that of the uninjected official base. Generalization on an adjacent monitoring domain improves significantly, and no monitored capability degrades. Mechanism experiments show that the gain depends on the semantic content of the distilled graph rather than its structural diversity alone does not produce it (directional evidence). With this framework and this regularity in hand, the use of LLM causal priors stops being manual trial and error and becomes an empirically verifiable selection problem.