Operationalising Weaker Faithfulness: Recovery, Ambiguity and Computational Limits

Cillian Hourican; Rick Quax

stat.MEOct 22, 2026

Operationalising Weaker Faithfulness: Recovery, Ambiguity and Computational Limits

Authors: Cillian Hourican, Rick Quax

Organizations: Computational Science Lab, Informatics Institute, University of Amsterdam. · Computational Science Lab, Informatics Institute, University of Amsterdam

Abstract

Variables can predict an outcome jointly while each is individually independent of it, causing structure learners to discard informative links. We investigate how to operationalise weaker faithfulness through two approaches: selective protection in an adjacency learner and explicit retention of alternative directed graph explanations. Matched experiments show that changes at both an early candidate-removal stage and later pruning can be necessary. In noisy XOR with a descendant (N = 10,000, 20 matched samples), combining reactivation and protection raises complete parent-pair recovery from 0/20 for either change alone to 20/20, while adding false edges. A confirmation study shows that the family estimator can preserve explanations lost by orienting the protected learner's selected skeleton: in one weaker signal XOR condition (N = 10,000) it retains the entire reference family in 48/50 samples, compared with 0/50 for the skeleton adapter. These gains are conditional. Protection can add only error, weaker effects produce false exclusions, and an ordinary faithful-chain comparison favours the PC learner. Complete nonempty family outputs decline at six/eight variables, and all 400 twelve/twenty-variable searches remain unfinished at the reporting cutoff. Neural controls show engineered retention, while the family estimator returns no accepted graph within its application budget. This operationalisation identifies two deletion stages that must be coordinated in the motivating example and shows when direct family search preserves explanations lost by selecting one skeleton. Finite evidence and computation remain substantial barriers to reliable general inference.

Figures & tables

Arm	Shrink	Protection	Global pruning
A	Original	Off	On
B	Reactivation	Off	On
C	Original	On	On
D	Reactivation	On	On
R	Reactivation	Off	Off

Table 1: Matched controls for protected adjacency learning. All arms use the same grow procedure and conditional-independence test (nominal level

.05

, conditioning cap two, no multiple-testing filtering). The stages are grow, shrink, edge assembly and global pruning; protection affects only the last stage. R is the no-pruning reference, outside the A–D factorial.

Panel of Figure 1 — Figure 1: Alternative explanations for independent fair-parent noisy XOR, under Markov compatibility, the declared weaker adjacency condition and subgraph minimality. The three DAGs have different missing adjacencies and collider roles. Orienting one skeleton cannot recover all three, and their edge union loses the joint constraints. This ambiguity is described by Marx et al. (2021, Section 7.1) . Time order supplies additional knowledge.

Object	Meaning
Population reference $\mathcal{F}^{*}$	All DAGs satisfying the stated restrictions for the known distribution.
Empirical reference $\mathcal{E}$	All DAGs satisfying those restrictions when evaluated against the complete table of finite CI decisions.
WF output	The result of the finite order search; even a completed output can differ from $\mathcal{E}$ .
Markov confidence family $\mathcal{C}$	A broader uncertainty construction without the weaker adjacency or minimality restrictions.
D adapter	Compatible orientations of the single skeleton selected by D.

Table 2: The reference families and estimated outputs answer different questions.

Figure 2: Operational choices affect both retention and errors. The matched stage comparison uses static noisy XOR with a descendant; the nuisance-variable comparison uses XOR with added independent variables. Both panels draw on the static controlled expansion, with 20 matched replicates per condition at

N=10{,}000

. Complete parent-pair recovery requires both designated parent links and differs from whole-skeleton recall. Uncertainty retains the 95% Wilson intervals for binary recovery and normal Monte Carlo intervals for mean false-edge counts. The no-pruning reference R is interpreted separately from the A–D factorial. The confirmation study uses separate samples.

Figure 3: Recovery, error and availability in four explanatory confirmation cells, all four-variable static inputs with

N=10{,}000

and 50 independent samples. All methods are compared with the same population WF reference in each cell. Coverage requires every reference DAG; equality also excludes extra DAGs. False assertions contradict at least one reference member and are scored only for complete nonempty outputs, with those denominators shown. Points and intervals retain saved 95% cellwise estimates. PC is included only in its eligible faithful-chain cell. The D adapter conditions on D’s selected skeleton and adds no CI evidence. Cells were selected after analysis for exposition; all planned cells remain in the full supplementary confirmation tables.

Variables	Completed / total inputs	Complete nonempty	Complete empty	Unfinished at cutoff
4	200/200	188	12	0
6	200/200	108	92	0
8	200/200	51	149	0
12	0/200	0	0	200
20	0/200	0	0	200

Table 3: Completion and output availability for the WF family estimator. Each row contains 200 confirmation inputs: ordinary and joint-effect networks, two parent settings and

N\in\{2{,}000,10{,}000\}

, with 25 inputs per cell. Counts span heterogeneous conditions and are not pooled accuracy estimates. Unfinished searches were discontinued at a cutoff chosen after computation began, without a common prespecified cumulative runtime limit. An empty family contains no accepted graph; an unfinished search has not established emptiness.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Condition	Definition
Symmetric XOR	Independent fair $X,Y$ ; $Z=X\oplus Y\oplus E$ , with the stated flip probability for $E$ ; independent $W\sim\mathrm{Bernoulli}(1/3)$ .
Asymmetric interaction	Independent $X\sim\mathrm{Bernoulli}(.35)$ , $Y\sim\mathrm{Bernoulli}(.60)$ ; $P(Z=1\mid X,Y)=(.10,.90,.80,.20)$ ; independent $W\sim\mathrm{Bernoulli}(1/3)$ .
Faithful chain	$X_{0}\sim\mathrm{Bernoulli}(1/3)$ and $X_{j}=X_{j-1}\oplus E_{j}$ for $j=1,2,3$ , with $.10$ flips.
Ordinary OR	Independent fair $X,Y$ ; $Z=(X\mathbin{\mathrm{OR}}Y)\oplus E$ , with $.10$ flips; independent $W\sim\mathrm{Bernoulli}(1/3)$ .
XOR with descendant	Independent fair $X,Y$ ; $Z=X\oplus Y\oplus E$ , $W=Z\oplus F$ , with $.10$ flips for both errors.
Independent-baseline episode	Baseline $X,Y$ are independent fair bits. Next slice: $Z=X\oplus Y\oplus E$ , $W=Y\oplus F$ , with $.10$ flips. Supplied tiers are $(0,0,1,1)$ for $(X,Y,Z,W)$ .

Appendix

Table 4: Mechanisms behind the reported confirmation examples. For a two-parent conditional-probability table, entries follow

00,01,10,11

Condition / $N$	Method	Covered	Equal	Any false	Nonempty	Mean size
XOR: flips .10 / 500	WF family	48/50	41/50	0/48	48/50	3.18
	D adapter	0/50	0/50	50/50	50/50	6.00
XOR: flips .10 / 2,000	WF family	42/50	38/50	2/44	44/50	2.74
	D adapter	0/50	0/50	50/50	50/50	6.00
XOR: flips .10 / 10,000	WF family	42/50	38/50	1/43	43/50	2.72
	D adapter	0/50	0/50	50/50	50/50	6.00

Appendix

Table 5: The four displayed family regimes at all planned sample sizes. Every row has 50 planned inputs. Coverage means retaining the entire population WF reference; equality means returning exactly that reference. Any false counts outputs with at least one false assertion, conditional on complete nonempty output; the actual denominator is shown. Mean family size includes complete empty outputs. All searches in these rows complete. PC is included only in its eligible faithful-chain cells. The underlying saved estimates and cellwise 95% intervals are supplied in family_metrics_source.csv , in the supplement’s figure_sources/ directory.

Type	Setting	Variables	$N$	Nonempty	Empty	Planted
J	1	4	2,000	23	2	0/25
J	1	4	10,000	25	0	1/25
J	2	4	2,000	18	7	0/25
J	2	4	10,000	25	0	1/25
O	1	4	2,000	25	0	24/25
O	1	4	10,000	23	2	22/25

Appendix

Table 6: Unpooled scaling accounting. Ordinary (O) and joint-effect (J) networks remain separate. Settings 1/2 use parent-selection probabilities .25/.65 and parent caps 2/4, respectively. Each row has 25 planned inputs. Complete nonempty, complete empty and incomplete outputs partition those inputs; planted membership includes complete empty failures. Partial counts incomplete searches with at least one accepted candidate DAG. A dash indicates no complete output. At four variables O is assumption-admitted and J is assumption-stress; above four, the full-joint reference and assumption classification are unavailable.

Profile	Bin (ms)	D/100	Exact	Recall	False edges	Exact bounds
XOR + descendant	5	49/100	0/49	1.000	1.714	$[0.00,0.51]$
	10	47/100	0/47	1.000	1.809	$[0.00,0.53]$
	20	49/100	2/49	1.000	1.592	$[0.02,0.53]$
Symmetric XOR	5	48/100	0/48	0.958	1.000	$[0.00,0.52]$
	10	49/100	0/49	0.939	1.020	$[0.00,0.51]$
	20	49/100	0/49	0.990	0.980	$[0.00,0.51]$

Appendix

Table 7: D’s local adjacency outcomes at the application reporting cutoff. Each profile/bin planned 100 synthetic randomizations on one fixed neural background. D/100 counts within-budget outputs; Exact counts outputs matching all known synthetic-incident adjacencies, including nonedges. Synthetic-incident edges have at least one synthetic endpoint, including links to the neural background. Recall and mean false-edge counts use the same evaluable outputs. Bounds retain all unattempted or limited cases and refer to all 100 planned inputs. Independent controls have no planted edges, so recall is undefined. These are local operational scores, without biological accuracy or an incremental D-versus-B comparison.

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