Coupling Noisy Pairwise Knowledge to the DAG Posterior for Causal Discovery

Guoliang Xu; James E Corter

cs.LGOct 3, 2026

Coupling Noisy Pairwise Knowledge to the DAG Posterior for Causal Discovery

Authors: Guoliang Xu, James E Corter

Organizations: Columbia University

Abstract

External causal reports can improve structure learning from limited observations, but their reliability varies across sources and variable pairs. We introduce HB-NoisyKG, a Bayesian framework that combines observational data with repeated causal reports from sources such as large language models. Each report is a noisy observation of a direct pair state implied by one DAG. A feature-conditioned Beta prior pools information about pair reliability, and a shared error matrix captures systematic mistakes. Alternating inference uses the graph posterior to refine reliability estimates, which determine how reports influence subsequent graph updates. The report likelihood uses only graph pair-state marginals, so the same observation layer supports discrete and continuous likelihoods in graph-only and joint inference. Against an 80-restart no-KG baseline, HB uses at most 80 total restarts and lowers mean SHD from 22.39 to 16.06 on five discrete benchmarks. On a physical light tunnel with random variable IDs and retained descriptions, HB lowers SHD from 39.00 for no-KG to 27.30. On continuous Sachs, graph-only BGe raises AUROC by 0.121 over no-KG Top-K. In a controlled synthetic study, continued updating also reduces mean reliability estimation error and held-out report log loss compared with one-time estimation.

Figures & tables

Figure 1: Matched report integration on five networks. Dots are paired SHD differences for 20 data seeds, averaging four report batches within each seed. Diamonds show means; bars are unadjusted 95% paired-bootstrap intervals, conditional on the benchmark graph and the four batches. Negative values favor HB. Panels use separate horizontal scales. All methods share the data and Top-

K

backend; HB and the fixed-channel control receive identical reports.

	No-KG		Fixed channel		Full HB
Network	20	80	20	80	$\leq 80$ total
Asia	$4.30$	$3.25$	$2.84$	$2.68$	$1.93$
Sachs	$12.60$	$11.20$	$7.04$	$6.68$	$7.24$
Child	$16.90$	$12.20$	$6.84$	$6.83$	$7.46$
Insurance	$42.45$	$40.10$	$35.59$	$31.23$	$34.61$
Alarm	$43.55$	$45.20$	$29.03$	$25.10$	$29.06$

Table 1: Mean SHD in the search-budget comparison. Each network uses 20 data seeds and four report batches. No-KG and the fixed channel use one graph call with the indicated restart count. Full HB uses 20 restarts per call, with at most four calls. All methods share the inputs of the reliability-assignment study; its full-HB fits provide the HB column.

Network	No-KG	HB	Fixed channel
Asia	$4.30$	$1.93$	$2.84$
Sachs	$12.80$	$7.25$	$7.05$
Child	$17.05$	$7.45$	$6.84$
Insurance	$42.45$	$34.49$	$35.49$
Alarm	$43.55$	$29.05$	$29.04$
Mean	$24.03$	$16.03$	$16.25$

Table 2: Matched direct-parent study (

N=1000

). (a) Structure recovery with canonical-name reports: HB and the fixed-channel control use identical reports and the same Top-

K

backend. SHD averages four source batches and 20 shared data seeds; reversals cost two. The mean weights networks equally. (b) Direct-edge F1 of aggregated LLM reports under three naming conditions, averaged over four source batches. The domain is fixed and variable descriptions are absent in all conditions.

		Natural labels		Random IDs
$N$	Model	SHD $\downarrow$	AUROC $\uparrow$	SHD $\downarrow$	AUROC $\uparrow$
1000	No-KG	$39.000$	$.7383$	$39.000$	$.7383$
	Common reliability	$44.800$	$.6961$	$42.500$	$.6992$
	Fixed- $\rho$ HB	$31.200$	$.7872$	$28.300$	$.8182$
	Full HB	$29.550$	$.7579$	$27.300$	$.7643$
250	No-KG	$47.900$	$.6938$	$47.900$	$.6938$

Table 3: Optical-system recovery. Fixed-

\rho

HB fixes reliability at

.8

and learns

B

. Both label conditions retain identical physical descriptions. Each report-based entry averages ten data blocks and four report batches. No-KG is fitted once per block and sample size, then reused across label conditions.

Inference	SHD	Reliability MAE	Held-out log loss
No-KG	$11.10$	–	–
Report-only estimation	$7.83$	$.246$	$.983$
One-time estimation	$7.00$	$.240$	$.874$
Full alternation	$6.63$	$.152$	$.762$

Table 4: Continued reliability estimation with synthetic reports. Means average three networks, four generating accuracies, and ten data seeds per condition. Reliability MAE averages

|\mathbb{E}_{q}\rho_{ij}-\alpha|

over pairs; log loss evaluates five independent held-out reports per pair. Lower is better in all columns.

Control	Change	Reused estimates	Purpose
No-KG	Omit report likelihood	None	Value of external reports
Fixed channel	Freeze a symmetric noise channel	None	Recovery with a supplied report channel
Fixed- $\rho$ HB	Fix $\rho=.8$ ; continue learning $B$	None	Reliability estimation within HB’s channel
Post-fit pooling	Replace pair weights by their observed-pair mean	Fitted reliability and $B$	Variation among learned pair weights
Common reliability	Fit one shared Beta reliability	None	Independent estimation of source trust
Pair-intercept	Fit pair reliabilities with an intercept-only prior	None	Alternative prior features and initialization

Table 5: Reliability controls and their purposes. Reused estimates are parameters obtained from a completed full-HB fit. The remaining controls are fitted independently.

	LLM reports		Homogeneous synthetic reports
Model	SHD $\downarrow$	AUROC $\uparrow$	SHD $\downarrow$	AUROC $\uparrow$
Common reliability	$19.92$	$.906$	$2.98$	$.950$
Pair-specific, intercept prior	$23.42$	$.873$	$4.52$	$.916$
Full HB, feature prior	$16.26$	$.916$	$4.94$	$.914$

Table 6: Independently fitted reliability models. All variants use one initialization and three graph updates. Left: equal-network means on the five-network LLM study. Right: equal-network and equal-accuracy means over seven synthetic accuracy levels, including

\alpha=0

. That endpoint drives the overall SHD difference: across the other six levels, mean SHD is

3.33

for HB and

3.35

for common reliability. These are separate refits from Table 2 .

(a) Variant	$\Delta$ SHD	$\Delta$ AUROC
Synthetic GT KG (oracle)	$-4.00$	$+0.019$
No confusion update	$+0.12$	$-0.006$
No $\beta,\tau$ update	$+0.20$	$-0.006$
No KG warm start	$+1.60$	$-0.053$
No local search	$+6.36$	$-0.028$

Table 7: Mechanism analysis. (a) Component shifts relative to HB-NoisyKG in the discrete benchmark suite. The matched fixed-channel comparison is in Table 2 (a). (b) Reliability separation, report-count effects on mean reliability estimation, and accurate-report summaries from the studies described in Appendix E .

Report setting	HB SHD	$\Delta$ SHD	HB AUROC
Clean	$2.80$	$-15.13$	$.969$
Missing, retention $.3$	$7.93$	$-10.00$	$.925$
Spurious, rate $.5$	$2.93$	$-15.00$	$.969$
Flipped, rate $.7$	$19.13$	$+1.20$	$.808$
Mixed, level $.9$	$19.80$	$+1.87$	$.830$

Table 8: Selected settings from the paired corruption study. No-KG has mean SHD

17.93

and AUROC

.880

. Differences are HB minus no-KG; lower SHD and higher AUROC are better. Complete settings are in Table 23 .

Method	AUROC $\uparrow$	E-SHD $\downarrow$	SHD $\downarrow$	$\bar{\rho}$
Continuous graph-only (BGe marginal likelihood)
MC3 (published)	$0.616\pm 0.027$	$34.0\pm 0.7$	–	–
BPC (published)	$0.566\pm 0.020$	$25.5\pm 2.3$	–	–
NOTEARS	$0.590\pm 0.012$	$18.10\pm 0.59$	$18.00\pm 1.33$	–
GOLEM	$0.586\pm 0.011$	$18.63\pm 0.74$	$19.00\pm 1.89$	–
BGES-style (continuous BGe)	$0.655\pm 0.000$	$15.00\pm 0.00$	$15.00\pm 0.00$	–

Table 9: Continuous Sachs results. Our rows use a shared condition-standardized protocol (mean

\pm

std over 20/10/10 restarts for graph-only / linear-joint / nonlinear-joint panels). MC3 and BPC are reproduced from Lorch et al. [22, Table 1] (30 restarts).

\bar{\rho}

is mean inferred reliability. DiBS+ coincides with DiBS in the joint panels.

$d$	SHD $\downarrow$	Seconds	RSS (GiB)
100	$101.1/53.2$	$1603/4668$	$.202$
125	$130.2/74.8$	$4089/12391$	$.254$

Table 10: Larger sparse binary DAGs: ten paired seeds,

N=1000

\alpha=.75

R=5

, 20 restarts per graph call and up to three HB updates. Entries give no-KG/HB means. RSS is measured HB memory.

Appendix figures & tables54 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Default	Description	Scope
$M$ (particles)	20	SVGD particle count	DiBS default
$T$ (steps)	2000	SVGD iterations	DiBS default
$K$ (restarts)	20	Restarts per graph call	Main discrete suite
$\tau$ (reliability prior)	8.0	Initial Beta concentration	HB default
$T_{\mathrm{EM}}$	$\leq 3$	Outer graph updates	Main discrete suite
Reward centering	mean	Remove state-independent offsets	HB default

Appendix

Table 11: Core hyperparameters and their scope. The outer-update limit and search budget are overridden by study as shown in Table 12 .

Study and arm	Restarts/call	Updates	Calls/fit	Warm blend
Benchmark suite, main HB	20	$\leq 3$	$\leq 4$	$1$
Suite secondary diagnostics	10	$\leq 2$	$\leq 3$	$1$
Matched naming, HB	20	$\leq 3$	$\leq 4$	$.5$
Matched naming, fixed channel/no-KG	20	–	$1$	–
Search-budget controls	$20/80$	–	$1$	–
Accuracy sweep, HB	20	$\leq 3$	$\leq 4$	$.1/.5/.9$

Appendix

Table 12: Top-

K

budgets by study. Calls include initialization. Limits permit early stopping; fixed update counts run to completion. Optical no-KG repeats the zero-reward call four times under the common budget.

Network	$d$	$\|E_{\text{true}}\|$	$\|\mathcal{K}\|$	Precision	Recall	F1
Asia	8	8	11	0.73	1.00	0.84
Sachs	11	17	12	0.58	0.41	0.48
Child	20	25	22	0.41	0.36	0.38
Insurance	27	52	37	0.51	0.37	0.43
Alarm	37	46	43	0.47	0.43	0.45

Appendix

Table 13: Directed-edge quality of the benchmark-suite causal-candidate reports.

|\mathcal{K}|

is the aggregated edge count. The four-batch naming intervention and direct/path prompt comparison use separate report sets.

Network	Direct prompt: adjacency F1	Path prompt: reachability F1
Asia	$0.570\pm 0.081$	$0.872\pm 0.021$
Sachs	$0.440\pm 0.037$	$0.429\pm 0.035$

Appendix

Table 14: Report quality under explicit causal targets, averaged over four source batches. Direct prompts are evaluated against adjacency; path prompts against reachability. Asia has 8 direct edges and 18 reachable ordered pairs.

Network	HB-direct	HB-path-aware	Fixed path-aware
Asia	$1.52$	$1.81$	$2.34$
Sachs	$7.86$	$8.05$	$8.03$

Appendix

Table 15: Inference from the same path reports. Each entry is mean SHD; lower is better. The fixed comparator uses a path-state observation model with

\rho=0.8

Network	Canonical	Anonymous IDs	Pseudo-scientific
Alarm	$0.355$	$0.000$	$0.228$
Asia	$0.591$	$0.000$	$0.000$
Child	$0.480$	$0.000$	$0.000$
Insurance	$0.401$	$0.000$	$0.000$
Sachs	$0.392$	$0.018$	$0.000$

Appendix

Table 16: Mean direct-edge F1 under the clean name-only intervention. Each value averages four source batches. A zero F1 can result from either absent predictions or incorrect predictions; the downstream comparison distinguishes these cases.

Network	Canonical	Anonymous IDs	Pseudo-scientific
Alarm	$30.00/0$	$0.00/20$	$14.50/6$
Asia	$9.75/0$	$0.50/17$	$0.00/20$
Child	$18.75/0$	$0.00/20$	$0.00/19$
Insurance	$26.50/0$	$0.00/19$	$0.00/17$
Sachs	$8.50/0$	$3.25/11$	$10.00/0$

Appendix

Table 17: Source coverage in the naming intervention: mean aggregated directional-edge count across four batches / number of empty rounds out of twenty.

Network	Labels	No-KG SHD	HB SHD	Fixed-channel SHD	HB $\bar{\rho}$
Asia	Canonical	$4.30$	$1.93$	$2.84$	$0.501$
	Anonymous	$4.30$	$3.95$	$4.49$	$0.403$
	Pseudo-scientific	$4.30$	$4.30$	$4.30$	$0.378$
Sachs	Canonical	$12.80$	$7.25$	$7.05$	$0.454$
	Anonymous	$12.80$	$13.31$	$13.43$	$0.434$
	Pseudo-scientific	$12.80$	$14.09$	$15.22$	$0.399$

Appendix

Table 18: Complete downstream naming comparisons. HB and the fixed-channel control use the same reports and graph backend, averaged over four batches and 20 shared data seeds. No-KG uses those datasets. The final column is HB’s mean inferred reliability.

Network	GPT-4.1-mini	GPT-5.1	GPT-4o	Claude Sonnet 4.5
Asia	$.783$	$.714$	$.800$	$.830$
Sachs	$.696$	$.684$	$.700$	$.640$
Child	$.804$	$.802$	$.799$	$.806$

Appendix

Table 19: Mean inferred reliability across LLM sources under the common source-study configuration. Each source has five report rounds and one sampled dataset per network.

Comparison	Metric	Mean difference
vs. DiBS-only	SHD	$-7.96$
vs. DiBS-only	AUROC	$+0.178$
vs. DiBS-only	neg. MLL	$-0.142$
vs. MMHC	SHD	$-2.44$
vs. MMHC	AUROC	$+0.222$
vs. MMHC	neg. MLL	$-0.184$

Appendix

Table 20: Aggregate differences in the discrete benchmark suite, reported as HB minus the comparison method over 25 network-seed evaluations. These are descriptive differences: seed evaluations reuse observations within each fixed network.

Variant	SHD $\downarrow$	AUROC $\uparrow$	neg. MLL $\downarrow$	$\bar{\rho}$
HB-NoisyKG (real multi-round LLM KG)	$14.84$	$0.925$	$9.624$	$0.425$
HB-NoisyKG (synthetic ground-truth KG)	$10.84$	$0.944$	$9.618$	$0.443$
HB-NoisyKG	$14.84$	$0.925$	$9.624$	$0.425$
No confusion update	$14.96$	$0.919$	$9.626$	$0.431$
No $\beta,\tau$ update	$15.04$	$0.919$	$9.629$	$0.440$
No KG warm start	$16.44$	$0.871$	$9.618$	$0.400$

Appendix

Table 21: Knowledge-source and component-ablation summaries for the main discrete benchmark suite.

Figure 3: Full reliability diagnostics for the main discrete benchmark suite. The learned reliability values rise with confidence and agreement, and correct claims receive higher $\rho$ than incorrect ones. The confusion panel summarizes the mean shared error structure.

Panel of Figure 4 — Figure 4: Discrete diagnostics. The sample-size sweep uses $N\in\{100,300,500,1000,5000\}$ . Negative MLL decreases across this sweep. Structural recovery is not monotone: SHD rises at $N=5000$ , while AUROC remains above its $N=100$ value. The ablation plot shows the effects of the individual components.

Corruption type	Severity range	Mean SHD (start $\to$ end)	Lowest mean AUROC	$\bar{\rho}$ range
Missing edges	retention $1.0\to 0.3$	$2.80\to 7.93$	$0.925$	$0.451$ – $0.483$
Spurious edges	spurious $0.0\to 0.5$	$2.80\to 2.93$	$0.969$	$0.483$ – $0.553$
Flipped directions	flip $0.0\to 0.7$	$2.80\to 19.13$	$0.808$	$0.418$ – $0.483$
Mixed corruption	mixed $0.1\to 0.9$	$3.73\to 19.80$	$0.830$	$0.487$ – $0.496$
No-KG	all settings	$17.93$	$0.880$	–

Appendix

Table 22: Fully paired corruption study, averaged over five networks and three sampling seeds per setting. SHD gives the means at the lowest and highest tested severity; AUROC gives the lowest severity-specific mean. Reliability ranges are across condition means, not individual runs.

Block	$r$	$s$	$f$	HB SHD	$\Delta$ SHD	$\Delta$ AUROC
Missing/spurious	$1$	$0$	$0$	$2.800$	$-15.133$	$+.089$
	$1$	$.2$	$0$	$2.867$	$-15.067$	$+.089$
	$1$	$.5$	$0$	$2.933$	$-15.000$	$+.089$
	$.7$	$0$	$0$	$3.467$	$-14.467$	$+.077$
	$.7$	$.2$	$0$	$3.467$	$-14.467$	$+.084$
	$.7$	$.5$	$0$	$3.467$	$-14.467$	$+.077$

Appendix

Table 23: Complete paired corruption profile.

r,s,f

denote retention, spurious-edge rate, and flip rate. No-KG has SHD

17.933

and AUROC

.880

in every row. Differences are HB minus no-KG; negative SHD and positive AUROC differences favor HB. The clean condition appears in both the missing/spurious and flip protocol blocks.

Figure 5: Fully paired corruption results. Grey points are the 15 network-seed evaluations per condition; blue curves are their means. Dashed lines show no-KG means for SHD and AUROC. Reliability averages all variable pairs within a run, including unreported pairs. The missing curve fixes spurious rate at zero, and the spurious curve fixes retention at one. Mixed-index settings are defined beside Table 22 .

Control	SHD $\downarrow$	E-SHD $\downarrow$	AUROC $\uparrow$	Brier $\downarrow$
No-KG	$23.960$	$24.149$	$.8965$	$.0614$
Full HB	$16.060$	$16.173$	$.9160$	$.0364$
Pooled Beta means	$16.265$	$16.391$	$.9220$	$.0335$
Pair-specific point means	$16.040$	$16.129$	$.9166$	$.0363$
Pooled observed-pair point mean	$16.265$	$16.391$	$.9210$	$.0337$
Network mean over all pairs	$21.075$	$21.224$	$.9046$	$.0428$

Appendix

Table 24: Reliability-assignment controls. Values are equal-network averages. Each report-based arm has 400 network–seed–batch conditions; no-KG is evaluated once per network and data seed. All edge scores use average-tie AUROC, and SHD counts reversals twice.

Network	Pair SHD	Pooled SHD	$\Delta$ SHD	95% CI	$p_{\mathrm{Holm}}$
Asia	$1.925$	$1.188$	$-.738$	$[-1.113,-.413]$	$.011$
Sachs	$7.338$	$7.338$	$.000$	$[-.513,.538]$	$1.000$
Child	$7.338$	$6.800$	$-.538$	$[-1.063,-.038]$	$.170$
Insurance	$34.475$	$36.713$	$2.238$	$[.500,3.950]$	$.096$
Alarm	$29.125$	$29.288$	$.163$	$[-1.075,1.275]$	$1.000$

Appendix

Table 25: Primary reliability-assignment comparison.

\Delta

SHD is pooled minus pair-specific point means, so positive values favor pair-specific weighting. Confidence intervals are unadjusted 95% bootstrap intervals for the mean difference.

Network	Common $\rho_{s}$	HB mean $\rho$	Common SHD	HB SHD	Initial no-KG
Asia	$.464$	$.638$	$1.80$	$1.93$	$4.30$
Sachs	$.714$	$.767$	$7.54$	$7.43$	$12.60$
Child	$.564$	$.684$	$6.71$	$7.61$	$16.90$
Insurance	$.219$	$.602$	$45.84$	$35.59$	$42.45$
Alarm	$.427$	$.657$	$37.73$	$28.73$	$43.55$

Appendix

Table 26: Learned trust and SHD in the independent-reliability comparison. HB reliability averages observed pairs; common reliability is a single source parameter. Initial no-KG is the shared one-call initialization. The batches and data seeds are those of Table 27 .

Network	Model	SHD	E-SHD	AUROC	Brier
Asia	Common	$1.800$	$1.994$	$.9124$	$.0310$
	Pair-int	$3.763$	$3.744$	$.8509$	$.0632$
	Full HB	$1.925$	$1.957$	$.9225$	$.0329$
Sachs	Common	$7.538$	$7.746$	$.9514$	$.0593$
	Pair-int	$13.275$	$13.559$	$.8769$	$.1139$
	Full HB	$7.425$	$7.536$	$.9442$	$.0608$

Appendix

Table 27: Independent reliability fits on canonical LLM reports. Each entry averages four source batches and twenty data seeds. Common means one jointly learned source reliability; Pair-int uses separate pair reliabilities with an intercept-only prior.

Network	$\Delta$ SHD	95% CI	$p_{\mathrm{Holm}}$
Asia	$.125$	$[-.263,.513]$	$1.000$
Sachs	$-.113$	$[-.750,.463]$	$1.000$
Child	$.900$	$[.250,1.575]$	$.066$
Insurance	$-10.250$	$[-13.338,-7.250]$	$.000048$
Alarm	$-9.000$	$[-12.075,-5.862]$	$.00027$

Appendix

Table 28: Primary matched-study contrast: full HB minus common reliability in SHD. Intervals are unadjusted 95% paired-bootstrap intervals over twenty data seeds after averaging source batches. Holm correction covers the five network tests.

Network	$\alpha$	Initial no-KG	Common	Pair-int	Full HB	$p_{\mathrm{Holm}}$
Asia	$0$	$5.5$	$.6$	$5.1$	$9.8$	$.027$
	$1/3$	$5.5$	$3.5$	$3.1$	$4.1$	$1.000$
	$.5$	$5.5$	$4.1$	$3.2$	$3.0$	$1.000$
	$.6$	$5.5$	$1.9$	$2.1$	$1.2$	$1.000$
	$.75$	$5.5$	$.8$	$1.4$	$1.4$	$1.000$
	$.9$	$5.5$	$0$	$1.1$	$.8$	$.094$

Appendix

Table 29: Independent reliability fits with a common generating accuracy across all pairs. Entries average ten seeds. Data-only values are the stored initializations, used as descriptive references; each fitted reliability variant then makes three additional graph calls.

$\alpha$	Estimation	SHD	Reliability MAE	Held-out log loss
$0$	Report-only	$18.13$	$.581$	$1.749$
	One-time	$14.97$	$.441$	$1.227$
	Full	$13.40$	$.259$	$.958$
$1/3$	Report-only	$10.43$	$.135$	$1.177$
	One-time	$9.37$	$.188$	$1.188$
	Full	$9.40$	$.127$	$1.159$

Appendix

Table 30: Feedback controls by report accuracy. Each row averages three networks and ten data seeds. Reliability MAE compares pairwise posterior means with generating accuracy. All held-out reports are excluded from fitting.

$\alpha$	Reused $R=5$	Intercept $R=5$	Disjoint $R=5$	Reused $R=10$
$0$	$13.40$	$7.53$	$12.07$	$10.83$
$1/3$	$9.40$	$8.57$	$8.57$	$7.53$
$.75$	$2.40$	$3.83$	$2.27$	$2.50$
$1$	$1.30$	$1.93$	$1.30$	$1.17$

Appendix

Table 31: Report-feature controls: mean SHD over three networks and ten seeds. Disjoint

R=5

has five likelihood reports and five auxiliary feature reports; reused

R=10

uses the same total report budget. No-KG has mean SHD

11.10

at every accuracy.

Network	No-KG	HB	Fixed channel $w=.99$	Hard use
Alarm	$46.20$	$2.50$	$1.63$	$0$
Asia	$5.00$	$.83$	$0$	$0$
Child	$16.43$	$3.17$	$1.57$	$0$
Insurance	$42.30$	$13.50$	$9.67$	$0$
Sachs	$12.10$	$1.07$	$.10$	$0$

Appendix

Table 32: Fully accurate positive-edge reports: mean SHD over thirty paired data seeds per network. The frozen channel uses

w=.99

\epsilon=.05

; hard use enforces the asserted edges. Unreported pairs abstain.

Figure 6: Reliability across accuracy and report count. Left: the seven-level accuracy sweep at

R=5

and

\lambda_{\mathrm{warm}}=0.5

; error bars are standard deviations across ten seeds. The dashed line marks exact calibration. Right: slopes of the fitted reliability–accuracy relationship in the report-count sweep.

	Slope				MAE
Network	$R=2$	$R=5$	$R=10$	$R=20$	$R=2$	$R=20$
Asia	$.265$	$.566$	$.747$	$.831$	$.197$	$.045$
Sachs	$.292$	$.595$	$.767$	$.848$	$.187$	$.043$

Appendix

Table 33: Report-count sweep. The first four columns give the slope of learned reliability against generating accuracy. The last two give calibration MAE at the smallest and largest report counts.

Error model	Asia $\Delta$ SHD	Sachs $\Delta$ SHD
Homogeneous, independent	$-4.5$	$-6.8$
Heterogeneous, independent	$-4.3$	$-5.1$
Homogeneous, correlated	$-3.4$	$-6.3$
Heterogeneous, correlated	$-2.6$	$-5.9$

Appendix

Table 34: Shared-

B

inference under heterogeneous and correlated reports. Differences are HB minus no-KG; negative SHD differences favor HB.

Figure 7: Sensitivity to abstention under independent and informative refusal. Each point is the mean paired SHD difference over ten seeds. Negative values favor HB; zero is the no-KG reference. All three mechanisms recover no-KG at 100% abstention.

Mechanism	$0\%$	$25\%$	$50\%$	$75\%$	$90\%$	$100\%$
Independent	$-4.1/-7.7$	$-4.1/-7.2$	$-3.8/-5.4$	$-2.6/-1.2$	$-1.3/+1.6$	$0/0$
Selective refusal	$-4.1/-7.7$	$-4.0/-7.9$	$-4.0/-7.9$	$-3.6/-5.8$	$-1.7/+.3$	$0/0$
Adverse refusal	$-4.1/-7.7$	$-3.8/-5.1$	$-2.8/-2.7$	$-.1/+1.3$	$+.9/+1.9$	$0/0$

Appendix

Table 35: Mean HB-minus-no-KG SHD under six abstention levels. Each entry gives Asia / Sachs. The curves are shown in Figure 7 .

	HB minus no-KG		HB minus fixed channel
Network	Difference [95% CI]	$p_{\mathrm{H}}$	Difference [95% CI]	$p_{\mathrm{H}}$
Asia	$-1.33\;[-1.78,-.86]$	$.0044$	$-.75\;[-1.10,-.36]$	$.0264$
Sachs	$-3.96\;[-5.45,-2.51]$	$.0039$	$.56\;[.00,1.14]$	$.0925$
Child	$-4.74\;[-6.68,-2.84]$	$.0054$	$.64\;[.04,1.20]$	$.0925$
Insurance	$-5.49\;[-8.89,-2.16]$	$.0280$	$3.39\;[1.51,5.46]$	$.0280$
Alarm	$-16.14\;[-19.94,-11.89]$	$.00019$	$3.96\;[1.66,6.43]$	$.0264$

Appendix

Table 36: HB minus each 80-restart control in SHD. Brackets give unadjusted 95% paired-bootstrap intervals;

p_{\mathrm{H}}

adjusts the ten network–comparator Wilcoxon tests. Negative differences favor HB.

Method and restarts	E-SHD $\downarrow$	AUROC $\uparrow$	Brier $\downarrow$
No-KG, 20	$24.1491$	$.8965$	$.06144$
No-KG, 80	$22.8228$	$.9065$	$.05088$
Fixed channel, 20	$16.4533$	$.9269$	$.03882$
Fixed channel, 80	$14.7927$	$.9336$	$.03552$
HB, $\leq 80$ total	$16.1730$	$.9160$	$.03642$

Appendix

Table 37: Secondary search-budget metrics, averaged equally over networks after averaging source batches within data seed. E-SHD uses ordered-adjacency error; AUROC gives half credit to tied scores.

Graph family	Method	AUROC $\uparrow$	E-SHD $\downarrow$	SHD $\downarrow$	$\bar{\rho}$
ER2-d20	DiBS	$0.956\pm 0.032$	$21.20\pm 11.17$	$21.2\pm 11.2$	–
ER2-d20	HB-NoisyKG	$0.941\pm 0.048$	$20.00\pm 15.28$	$20.0\pm 15.3$	$0.400$
ER4-d20	DiBS	$0.834\pm 0.071$	$51.20\pm 18.50$	$51.2\pm 18.5$	–
ER4-d20	HB-NoisyKG	$\mathbf{0.843\pm 0.052}$	$\mathbf{42.60\pm 19.86}$	$\mathbf{42.6\pm 19.9}$	$0.405$

Appendix

Table 38: Small synthetic continuous graph-only check on two

d{=}20

Erdős–Rényi linear-Gaussian families with controlled noisy KGs. All values are mean

\pm

std over 5 repeats.

		SHD $\downarrow$		E-SHD $\downarrow$
SEM	$d$	DiBS	HB	DiBS	HB
Linear	$20$	$21.96$	$20.57$	$21.96$	$20.57$
Linear	$30$	$48.59$	$45.20$	$48.59$	$45.20$
Nonlinear	$20$	$26.12$	$21.41$	$26.12$	$21.42$
Nonlinear	$30$	$43.08$	$41.00$	$43.08$	$40.99$

Appendix

Table 39: Structural recovery in the synthetic ER comparison. Means cover five graphs, five data seeds per graph, and three report accuracies per dataset. Both metrics are lower-is-better. Repeated conditions are averaged within graphs for the analysis in Appendix F.1 , with the base priors and total graph-update budgets specified below.

SEM	$d$	Metric	Mean difference	$p$	$p_{H}$
Linear	20	E-SHD	$-1.39$	$.375$	$.500$
Linear	30	E-SHD	$-3.39$	$.0625$	$.250$
Nonlinear	20	E-SHD	$-4.70$	$.0625$	$.250$
Nonlinear	30	E-SHD	$-2.09$	$.250$	$.500$

Appendix

Table 40: Graph-level E2 structural analysis. Differences are HB minus DiBS. Individual graph effects appear in Table 41 ;

p

is the exact sign-flip value and

p_{H}

is adjusted across four structural tests.

SEM	$d$	Metric	Graph 1	Graph 2	Graph 3	Graph 4	Graph 5
Linear	20	E-SHD	$2.53$	$-4.53$	$.73$	$-2.33$	$-3.33$
Linear	30	E-SHD	$-1.20$	$-4.53$	$-2.67$	$-1.60$	$-6.93$
Nonlinear	20	E-SHD	$-5.47$	$-4.80$	$-1.87$	$-6.20$	$-5.18$
Nonlinear	30	E-SHD	$2.53$	$-6.39$	$-3.73$	$-1.20$	$-1.67$

Appendix

Table 41: Individual graph effects underlying Table 40 . Each graph effect averages five data seeds and three report accuracies.

Method	$d=20$	$d=30$
NOTEARS	$.60$	$1.64$
GOLEM	$3.68$	$9.96$
KG-SoftMAP (discretized BDeu)	$7.43$	$14.80$
HB-NoisyKG	$20.57$	$45.20$
DiBS-joint	$21.96$	$48.59$

Appendix

Table 42: Linear E2 structural comparisons across all evaluated methods. Each entry is mean SHD over 75 conditions based on 25 observational datasets and five graph draws. Method implementations are described in the text.

$d$	$\alpha$	$n$	$\bar{\rho}$ (mean $\pm$ std)
$20$	$0.60$	$25$	$0.523\pm 0.035$
$20$	$0.75$	$25$	$0.630\pm 0.031$
$20$	$0.90$	$25$	$0.736\pm 0.023$
$30$	$0.60$	$25$	$0.529\pm 0.010$
$30$	$0.75$	$25$	$0.636\pm 0.015$
$30$	$0.90$	$25$	$0.750\pm 0.018$

Appendix

Table 43: Learned reliability in the nonlinear E2 study. Each row averages five graphs and five data seeds. Report accuracy and reported confidence both equal

\alpha

in this generator; Appendix E.5 supplies a separate controlled-report study.

$d$	$\alpha$	$n$	$\Delta$ E-SHD $\downarrow$	$p_{\Delta\mathrm{E-SHD}}$	$\bar{\rho}$
$20$	$0.60$	$25$	$-4.59$	$.0625$	$0.523$
$20$	$0.75$	$25$	$-4.44$	$.0625$	$0.630$
$20$	$0.90$	$25$	$-5.08$	$.0625$	$0.736$
$30$	$0.60$	$25$	$-0.80$	$.750$	$0.529$
$30$	$0.75$	$25$	$-2.72$	$.0625$	$0.636$
$30$	$0.90$	$25$	$-2.76$	$.125$	$0.750$

Appendix

Table 44: Nonlinear E2 results by report accuracy. Differences are HB minus DiBS. Each row includes 25 datasets from five graphs;

p

-values are raw two-sided graph-level sign-flip values after averaging the five seeds within each graph.

\bar{\rho}

is mean learned reliability.

$d$	$\alpha$	$n$	$\Delta$ E-SHD $\downarrow$	$p_{\Delta\mathrm{E-SHD}}$	$\bar{\rho}$
$50$	$0.60$	$25$	$-3.16$	$.1875$	$0.576$
$50$	$0.75$	$25$	$-1.04$	$.5625$	$0.704$
$50$	$0.90$	$25$	$-6.51$	$.250$	$0.781$
Pooled (75 conditions)		$75$	$-3.57$	$.125$	—

Appendix

Table 45: Linear

d=50

extension. Differences are HB minus DiBS. There are 25 datasets per accuracy from five graphs. Raw

p

-values use the graph-level two-sided sign-flip test; the pooled row first averages accuracies and seeds within each graph.

Family	Particles/steps	DiBS	HB	$\Delta$ E-SHD
Linear	$20/2{,}000$	$20.0$	$26.0$	$+6.0$
Linear	$40/4{,}000$	$18.4$	$15.0$	$-3.4$
Nonlinear	$20/2{,}000$	$22.6$	$18.4$	$-4.2$
Nonlinear	$40/4{,}000$	$19.6$	$14.6$	$-5.0$

Appendix

Table 46: Joint inference at two per-call SVGD budgets. Each result is mean E-SHD over five seeds. The last column gives HB minus DiBS differences. The two budgets use the same selected graph, observations, reports, and seeds.

Family	No-KG	HB	Fixed channel $w=.99$
Linear	$20.9$	$21.4$	$17.0$
Nonlinear	$22.7$	$22.4$	$21.2$

Appendix

Table 47: Joint inference with accurate reports. Entries are mean E-SHD over ten paired runs. All methods use 40 particles and 4,000 steps per graph call.

$B^{\star}$ structure	$R$	$\mathrm{MAE}(\hat{\rho}-\rho^{\star})$ $\downarrow$	$\\|\hat{B}-B^{\star}\\|_{F}$ $\downarrow$
symmetric	$3$	$0.192\pm 0.006$	$0.280\pm 0.121$
symmetric	$5$	$0.167\pm 0.006$	$0.196\pm 0.114$
symmetric	$10$	$\mathbf{0.128\pm 0.007}$	$\mathbf{0.173\pm 0.091}$
asymmetric	$3$	$0.194\pm 0.006$	$0.249\pm 0.137$
asymmetric	$5$	$0.168\pm 0.007$	$0.216\pm 0.111$
asymmetric	$10$	$\mathbf{0.127\pm 0.006}$	$\mathbf{0.163\pm 0.090}$

Appendix

Table 48:

G

-known recovery of

\rho

and

B

as a function of round budget

R

. All values are mean

\pm

std over

50

Monte Carlo seeds per

(R,B^{\star})

setting. The symmetric and asymmetric rows differ only in

B^{\star}

; the ground-truth DAG, node labels, and

\rho^{\star}

population are identical across rows.

$\beta_{\mathrm{var}}$	$\tau_{\mathrm{shape}}$	$\lambda_{\mathrm{warm}}$	Asia SHD $\downarrow$	Sachs SHD $\downarrow$
1	1	0.1	6.3	11.0
1	1	0.5	9.2	16.6
1	1	0.9	10.9	17.3
1	2	0.1	6.7	11.2
1	2	0.5	9.2	16.6
1	2	0.9	10.9	16.7

Appendix

Table 49: Hyperparameter sensitivity. Each row averages ten seeds. Most variation is associated with the warm strength

\lambda_{\mathrm{warm}}

; at a fixed strength, changes in the hyperprior parameters have smaller effects.

		Final $\bar{\rho}$ (HB)			SHD $\downarrow$
Network	$\alpha$	$.1$	$.5$	$.9$	No-KG	Fixed channel	HB $.1$	HB $.5$	HB $.9$
Asia	$0$	$.260$	$.279$	$.326$	$5.5$	$21.3$	$8.4$	$10.0$	$12.7$
	$1/3$	$.442$	$.446$	$.449$	$5.5$	$7.3$	$4.1$	$4.4$	$4.1$
	$.50$	$.545$	$.548$	$.550$	$5.5$	$4.1$	$3.0$	$3.0$	$3.9$
	$.60$	$.614$	$.614$	$.621$	$5.5$	$.5$	$1.7$	$1.2$	$.8$
	$.75$	$.701$	$.704$	$.711$	$5.5$	$.9$	$1.5$	$1.4$	$1.1$

Appendix

Table 50: Accuracy and warm-strength study (

N=1000

R=5

). Each entry averages ten paired seeds. HB columns are indexed by

\lambda_{\mathrm{warm}}

; no-KG and fixed-channel SHD provide reference comparisons. The fixed channel uses

w=.8

\epsilon=.05

. Observations are held fixed across accuracies, so no-KG values repeat across rows.

Rule	Asia $\Delta$ SHD	Sachs $\Delta$ SHD
Data/report conflict	$-.03$	$+.83$
Posterior predictive	$+.03$	$+.14$

Appendix

Table 51: Warm selection relative to fixed

\lambda_{\mathrm{warm}}=.5

. Entries are mean SHD differences; negative values favor the selector.

Figure 9: Empirical no-KG and HB runtime on the five discrete benchmarks. Axes are logarithmic, and each point is labeled by its network. The curves use the same graph-search protocol within each comparison.

Backend	Cells	Period-two	Joint tolerance	Boundary flags
Exact posterior	3,240	0	$96.6\%$	2,143
Top- $K$	2,160	2	$81.8\%$	2,140
DiBS-SVGD	108	0	$9/108$	108

Appendix

Table 52: Inference dynamics over fifty updates. Non-finite values and period-two events measure numerical failure and oscillation. Boundary flags measure concentration. Joint tolerance records attainment of the strict joint-state criterion and is sensitive to Monte Carlo variation in DiBS-SVGD.

Backend	$\alpha$	Cells	Either	SHD	AUROC	$\Delta$ AUROC	$\Delta$ Brier
Exact	$0$	1,080	610	12	610	$-.1499$	$+.1027$
	$1/3$	1,080	561	0	561	$-.1239$	$+.0335$
	$.75$	1,080	485	0	485	$-.0840$	$-.0200$
Top- $K$	$0$	720	474	289	457	$-.1468$	$+.0852$
	$1/3$	720	323	153	275	$-.0743$	$+.0361$
	$.75$	720	90	8	89	$+.0084$	$-.0210$

Appendix

Table 53: Performance-loss flags for Exact and Top-

K

inference by report accuracy. “Either” counts the union of the unchanged SHD and AUROC criteria. The final columns give mean HB-minus-no-KG AUROC and Brier differences. AUROC uses standard handling of tied scores. Counts refer to this stress-test grid.

Method	True-class mass $\uparrow$	Pair-state error $\downarrow$
No-KG	$.889$	$.045$
HB, $\alpha=.60$	$.933$	$.028$
HB, $\alpha=.75$	$.925$	$.031$
HB, $\alpha=.90$	$.967$	$.015$

Appendix

Table 54: Controlled CPDAG study. Higher posterior class mass and lower expected pair-state error indicate better recovery. Values average the tested motifs, sample sizes, and seeds.

Graph class	Quantity	No-KG	HB
ADMG	True-candidate mass $\uparrow$	$.197$	$.812$
ADMG	Mass on $0\leftrightarrow 1$ $\uparrow$	$.591$	$.926$
Cyclic	SHD $\downarrow$	$3.03$	$.57$

Appendix

Table 55: Controlled tests with compatible graph support at

\alpha=.75

. Higher posterior masses and lower SHD are better. These tests concern finite four-node candidate sets.

Labels	Reports	Precision	Recall	F1	Coverage	Edges
Natural	Single round	$.628$	$.267$	$.374$	$.086$	$16.40$
Natural	At least $2/5$ rounds	$.629$	$.301$	$.407$	$.099$	$18.75$
Random IDs	Single round	$.669$	$.282$	$.397$	$.087$	$16.45$
Random IDs	At least $2/5$ rounds	$.657$	$.282$	$.395$	$.088$	$16.75$

Appendix

Table 56: LLM report quality on the optical task. Coverage concerns unordered pairs; precision, recall, and F1 concern directed edges.

$N$	Labels	Model	SHD	E-SHD	AUROC	Brier
1000	Natural	No-KG	$39.000$	$39.9274$	$.7383$	$.0945$
		HB	$29.550$	$29.5899$	$.7579$	$.0765$
		Fixed- $\rho$ HB	$31.200$	$31.3108$	$.7872$	$.0814$
		Common reliability	$44.800$	$44.8140$	$.6961$	$.1141$
	Random IDs	No-KG	$39.000$	$39.9274$	$.7383$	$.0945$
		HB	$27.300$	$27.2371$	$.7643$	$.0702$

Appendix

Table 57: All optical-task metrics. Fixed-

\rho

HB uses

\rho=.8

with learned

B

. Report-based means average ten blocks and four report batches; the repeated no-KG entries denote the same ten fits at each sample size. SHD, E-SHD, and Brier are lower-is-better; AUROC is higher-is-better.

$N$	Labels	Comparator	$\Delta$ SHD [95% interval]	$\Delta$ AUROC [95% interval]
1000	Natural	No-KG	$-9.45\;[-16.30,-1.38]$	$.0195\;[.0044,.0336]$
		Fixed- $\rho$ HB	$-1.65\;[-5.48,1.73]$	$-.0294\;[-.0573,-.0026]$
		Common reliability	$-15.25\;[-28.60,-3.40]$	$.0617\;[.0260,.0968]$
	Random IDs	No-KG	$-11.70\;[-16.68,-6.43]$	$.0260\;[.0088,.0419]$
		Fixed- $\rho$ HB	$-1.00\;[-6.73,4.40]$	$-.0538\;[-.0896,-.0159]$
		Common reliability	$-15.20\;[-28.13,-4.13]$	$.0651\;[.0213,.1095]$

Appendix

Table 58: Optical-task paired contrasts: HB minus the comparator, with exploratory 95% crossed-bootstrap intervals. Fixed-

\rho

HB uses

\rho=.8

with learned

B

. Negative SHD and positive AUROC differences favor HB.

Causal Discovery

Causal discovery infers cause-and-effect structures from observational and interventional data, distinguishing causal relationships from correlation.

See all 222 papers

Jul 11, 2026

Integrating Background Knowledge for Scalable Causal Discovery

Expert background knowledge is often available in practical applications of causal discovery. Such constraints on the true causal graph can help causal discovery in terms of identifiability of causal effects and accuracy of the learned structure, but also in reducing the space of candidate causal graphs. As causal discovery can become computationally expensive for large number of variables, it is crucial to utilize background knowledge effectively during the causal discovery process. However, most current methods only use background knowledge in a postprocessing step after causal discovery to refine the learned graph. In this work, we develop a framework for utilizing background knowledge during the causal discovery process, focusing especially on scalable causal discovery methods that recover only a subset of the whole graph. We implement our framework for multiple algorithms and empirically show that utilizing background knowledge can both reduce computational requirements and increase the quality of the learned structures.

Mátyás Schubert, Theofanis Aslanidis, Tom Claassen +1

University of Amsterdam · Radboud University Nijmegen · Saarland University

Causal Discovery Causal Structure Learning

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May 26, 2026

Iterative Causal Discovery: Per-Edge Impossibility Certificates, Tier-Aware Oracle Queries, and the $1+K$ Lower Bound

Causal-discovery algorithms return a directed graph, yet provide no principled means of distinguishing edge directions identified by the data from those assigned without an identifying assumption. Under the standard Markov and faithfulness conditions, the observational distribution identifies only a Markov equivalence class; orientations within that class are not determined by the joint distribution and cannot be recovered from additional samples alone, but require either a functional restriction or an intervention. We introduce a protocol for observational causal discovery on continuous data that attaches to each candidate edge a discrete impossibility certificate: a RESOLVED code records the identifiability theorem under which the direction was committed, while an IMPOSSIBLE code records the failure mode together with the specific question a domain expert must answer to resolve it. The bivariate cascade is extended with five gated identifiability tiers LSNM, IGCI, Stein, MDL, and PEIT that abstain when their precondition test rejects. Two oracle primitives, the meta-hub query and the node-children query, jointly establish an upper bound of

1+K

expert interactions sufficient to recover any DAG, where

K

denotes the number of non-leaf vertices. Under an ideal-oracle assumption, the bound is met exactly on the asia, sachs, child, and alarm benchmarks.

Eichi Uehara

aflo, Inc., Tokyo, Japan

Bayesian Networks Causal Discovery Causal Structure Learning Causal Identifiability

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Aug 5, 2026

SVI-DAG: A Structured Variational Inference Approach to Bayesian Causal Discovery

Bayesian causal discovery seeks to determine the posterior distribution of causal theories, which are interpreted as directed acyclic graphs (DAGs) that explain the observed data. The resulting posterior allows systematic reasoning regarding epistemic uncertainty within these theories. Nonetheless, finding such graphs is difficult due to identifiability problems and limited observational data. Furthermore, precisely approximating posterior over graphs is challenging given vast range of potential DAGs. Recent Bayesian approaches have addressed some of these challenges, yet they remain limited as they fail to encode dependencies between edges, and lack principled ways to incorporate domain knowledge as inductive biases during the search process. To overcome these limitations, we propose SVI-DAG, a structured variational inference approach to Bayesian causal discovery using observational data and prior beliefs that uses normalizing flows to model dependencies between edges, supporting expressive and multimodal posterior learning over DAGs. To mitigate mode seeking behaviour in evidence lower bound optimization and promote mode coverage, we use stein variational gradient descent to update the node potentials using a kernel in acyclicity space. We evaluate SVI-DAG against 5 state-of-the-art Bayesian DAG learning methods and demonstrate superior performance in uncertainty quantification while remaining competitive in terms of structural accuracy.

Shrenik Zinage

Massachusetts Institute of Technology · This work was conducted while the author was a PhD student at Purdue University, independently of the author’s dissertation research.

Causal Discovery Variational Inference

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Abstract

Causal Discovery

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