When and Why LLM Causal Priors Help: Closed-Loop Prior Selection for Amortized Causal Inference

Haohao Zhou

cs.AISep 7, 2026

When and Why LLM Causal Priors Help: Closed-Loop Prior Selection for Amortized Causal Inference

Authors: Haohao Zhou

Organizations: National Key Laboratory of Information Systems Engineering National University of Defense Technology, Changsha, Hunan, China

Abstract

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 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.

Figures & tables

Figure 1: Schematic of the closed-loop prior selection framework (two-stage loop, Algorithm 1 ). Conceptual schematic.

Module	Question answered	Key result	Statistical level
M1 (§ 5.1 )	Content or narrowing?	$\sim$ 72% narrowing, $\sim$ 28% content	Content share directional ( $n=3$ , $p=0.071$ )
M2 (§ 5.2 )	Can gain and collateral cost be decoupled?	Non-monotone interior optimum; adjacent domain degrades monotonically	$n=3$ descriptive
M3 (§ 5.3 )	Can multiple sources be combined?	Ensembling judged negative, degradation of $0.111$	Directional ( $p=0.171$ )
M4 (§ 5.4 )	Which one to inject; can it be automated?	Winner $2.75\times$ gain, three improvements	Formally significant ( $p=0.0086$ , $p=0.0109$ , $n=5$ )
M5 (§ 5.5 )	Why does the gain occur?	Content beats diversity, V-shaped temperature	Directional ( $n=3$ )
M7 (§ 5.6 )	Does it transfer to a new domain?	sales from-scratch $3.0\times$ gain, does not reach official	Directional ( $n=3$ , unpaired, $p=0.0001$ )

Table 1: Module map: the question each experiment answers, the key result, and the statistical level.

Figure 2: M1 single-source injection gain: primary-domain NMSE of the five configurations ( $n=3$ , error bars $\pm 1$ sample std); dashed line is the pre-registered competence threshold $0.062$ . The distilled graph is statistically indistinguishable from the true graph and better than the general prior and random graph (directional at $n=3$ ).

Figure 3: M2 observational-regime ratio curve ( $n=3$ ): the primary domain shows a non-monotone interior optimum (50/30/20 mix best, first crossing the competence threshold $0.062$ ); the adjacent domain degrades monotonically along the same axis. Both axes are NMSE.

Figure 4: M4 main result: mixA and the winner configuration paired strictly by seed ( $n=5$ , seed $1\leftrightarrow 1$ through $5\leftrightarrow 5$ ), five seed-connection lines all sloping downward; squares are means. The horizontal dashed line is the uninjected official base $0.0308$ , reference only and not part of the paired test. Paired $t=4.80$ , $p=0.0086$ , a $2.75\times$ gain, direction 5/5.

Dimension	Winner v4t15	Control	Conclusion
law_race (primary)	$0.0211\pm 0.0048$	mixA $0.0580\pm 0.0143$ ; official base $0.0308\pm 0.0083$	$2.75\times$ gain, formally significant ( $p=0.0086$ ); error level below the official base (descriptive cross-lineage, see text)
sales (adjacent monitor)	$0.356\pm 0.020$	mixA $0.389\pm 0.013$	Not broken but improved, formally significant ( $p=0.0109$ , direction 5/5)
Diagnostic panel (general)	$0.0092$ – $0.0120$	Base level $\sim 0.010$	No observed forgetting
Cross-seed std	$0.0048$	$0.0143$	About $3\times$ narrower

Table 2: Full set of main-result metrics (

n=5

\pm 1

sample std; significance criterion

p<0.05

and

|\Delta|\geq 0.02

Figure 5: Cheap screening (5k-step NMSE) versus full training (20k-step NMSE) for the three candidates with paired data; dashed lines mark the mixA full-training reference $0.0580$ ; the dotted line is $y=x$ . The random graph ranks second by cheap screening yet reverses under full training to below the baseline.

Figure 6: M5 mechanism: temperature-axis three-point line ( $n=3$ , no interpolation); $T=0.7$ degrades due to edge-drop narrowing ( $p=0.030$ , directional under the $n=3$ discipline); the hollow point is the random-graph control (same structural diversity, shuffled semantics), $2.1\times$ worse than the baseline (directional, $p=0.093$ ), rejecting the “pure diversity” hypothesis.

Figure 7: M7 cross-domain closed loop: sales-specific prior injection ( $n=3$ ) versus the law_race prior ( $n=3$ ), per-seed dots plus means; the horizontal dashed line is the official base’s reference value $0.0336$ on sales (not reached, recorded honestly). Unpaired $t=-15.21$ , $p=0.0001$ , a $3.0\times$ gain (directional at $n=3$ ).

Figure 8: M6 base dependence: the same recipe injected into two bases, before/after comparison ( $n=3$ ). The v2 base improves substantially ( $0.221\to 0.021$ ); the official base degrades about $9\times$ ( $0.0308\to 0.279$ ), and the ranking of the two configurations fully reverses relative to M4.

Method	law_race NMSE	sales NMSE	Outcome vs. baselines
Do-PFN v4t15 (injected)	$\mathbf{0.0211}$	$0.3562$	On law_race, leads all seven baselines in point estimate; significantly better than all machine-learning baselines ( $p\leq 0.0060$ vs CausalPFN; $p\leq 0.0017$ vs S/T/X/DR-learner and CausalForestDML); no significant difference from naive ATE. On sales, last; significantly worse than all six baselines ( $p\leq 0.030$ ).
Do-PFN official	$0.0308$	$0.0336$	On law_race, second; significantly better than CausalPFN ( $p=0.0028$ ) and all four meta/forest baselines; no significant difference from naive ATE ( $p=0.569$ ). On sales, fifth; directional losses to the top four ( $p\in[0.051,0.079]$ ); significantly better than S-learner only ( $p=0.022$ ).
CausalPFN	$0.1276\pm 0.0339$	$0.1019\pm 0.0671$	Direct SOTA.
naive ATE	$0.0278$	$\mathbf{0.0103}$	Constant-effect baseline; top on sales (effect near-constant).
DR-learner	$0.7311$	$0.0103$
T-learner	$0.2708$	$0.0109$

Table 3: M8 full comparison on our two domains (

n=5

fixed splits, paired by split; NMSE, lower is better). Rows are sorted by law_race NMSE. Per-pair

p

-values are paired

t

-tests of each Do-PFN arm against the named baseline.

Figure 9: M8 SOTA comparison (same 5-fold protocol): left panel, on law_race both Do-PFN versions lead all machine-learning baselines in point estimate ( $4.1\times$ / $6.1\times$ over CausalPFN, $p\leq 0.006$ ); right panel, on sales simple meta-learners beat all amortized models. naive ATE is the constant baseline. Per-pair $p$ -values in Table 3 .

Figure 10: M9 standard benchmarks (the two panels have different metric units and are plotted separately): left panel, IHDP PEHE ( $n=50$ realization pairing, effect-size threshold $|\Delta|\geq 0.05$ ), CausalPFN in a class of its own ( $0.7444$ ), the three Do-PFN versions in $4.87$ – $6.96$ , the injected configuration bottom; right panel, Lalonde ATE relative deviation ( $n=5$ ), the two uninjected bases on par with classical methods statistically, the injected configuration significantly worse than 6/7 baselines.

Figure 11: Conceptual schematic of the four diagnosis–generalization divergence instances placed in the (probe direction, transfer direction) plane; the reverse quadrants (probe worse but transfer better, and probe better but transfer worse) are shaded. Conceptual schematic, not per-instance measured coordinates.

Condition	Supporting instance	Evidence	Limitation
Cond. 1 base underfit	v2 base law_race (M4); v2 base sales (M7)	$2.75\times$ significant (M4, $n=5$ ); $3.0\times$ directional (M7, $n=3$ )	Counterexample is M6 official-base degradation; only two bases, and no protocol re-search on the official base
Cond. 2 prior-domain match	law_race prior used on IHDP, Lalonde (M9 stage 1)	Significantly negative contribution on both domains	Only negative instances of domain mismatch; no positive replication of domain match
Cond. 3 task within support	IHDP 26 dim $\gg$ $[1,6]$ (M9 stage 2)	Domain match still does not repair	Single instance; attribution not separated
All three hold	v2 base law_race (M4)	Significant gain	Only one complete condition-combination cell

Table 4: Evidence supporting the three-condition empirical regularity.

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