Causal effect estimation asks how an outcome would change under an intervention and is foundational for medicine, economics, and public policy. Prior‑data fitted networks (PFNs) amortize this task: a model trained on massive programmatically generated synthetic causal tasks reads a new problem's observational data as context and returns an interventional effect estimate in a single forward pass. The model's capability is largely determined by the synthetic training prior, which is still handcrafted—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. We investigate whether injecting such graphs helps, where any gain originates, and when injection is beneficial. 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 top‑scoring prior then receives full training and paired statistical validation.
On a 7.34M‑parameter Do‑PFN, the framework's winner achieves a formally significant $2.75\times$ gain on the primary evaluation domain, with error below that of the uninjected official base. Generalization on an adjacent monitoring domain improves markedly, and no monitored capability degrades. Mechanism experiments show that the gain depends on the semantic content of the distilled graph rather than on structural diversity alone (directional evidence). With this framework and regularity in hand, the use of LLM causal priors ceases to be manual trial‑and‑error and becomes an empirically verifiable selection problem.
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