A Computationally Feasible Framework for Causal Probabilistic Explanation

• In the OBCB example, witness mechanisms make PCI rank gender above credit, correcting attribution methods that split responsibility despite the credit check never being active.Witnesses can also widen a mediator–upstream-cause margin from 0.034 to 0.133 and distinguish reliable from unreliable causes across forensic scenarios.Source: §1 · pp. 3, 6
• PCI addresses actual causality’s all-or-nothing, computational, and deterministic-model limitations by producing graded scores for large stochastic, discrete, continuous, and machine-learned causal models.The method targets tractable estimation without exhaustive enumeration, although rare dependencies can still be missed or require sampling effort comparable to enumeration.Source: §1 · pp. 3, 4, 6
• PCI combines actual causality’s context sensitivity with probability-of-causation methods, replacing exhaustive witness-set search with an expectation over Γ and Monte Carlo estimation.It uses sufficiency and necessity counterfactual checks, an alternative-value distribution ∆, and a causal impact function, while requiring a trained model capable of counterfactual sampling.Source: §1 · pp. 3, 5
• PCI’s empirical preview reports agreement with actual-causality verdicts across ten synthetic checks and tractability beyond exact enumeration’s feasible problem sizes under a 60-minute budget.The preview also reports continuing from 17 variables to roughly 73–145 variables, with a 0.85 correct-attribution rate through 89 variables.Source: Figure 1 · p. 4, §1 · p. 7
• Under specified choices of Γ and ∆, PCI assigns a positive score exactly when a candidate is an actual cause in Halpern’s sense.This gives approximate actual-causality verdicts without enumerating witnesses, while preserving the possibility of graded responsibility more generally.Source: §1 · pp. 6–7
• PCI presumes a fully specified probabilistic causal model, and its posterior score reflects uncertainty over mechanisms and parameters rather than solving model non-identification.For complex neural or Gaussian-process mechanisms, obtaining a data-responsive posterior remains a separate problem; posterior spread can instead support sensitivity analysis.Source: §1 · pp. 6–7

2 Causal Impact: Motivations

• Causal attribution must identify which counterfactuals matter for a specific outcome, because a generally causal feature can be inactive in the case at hand.Type-level causal strength answers whether X tends to cause Y, whereas token-level actual causation asks whether X’s contribution was live in the specific situation.Source: §2 · pp. 8–9

2.1 Actual Causality

• In Alice’s denial, gender alone is responsible because it preempts the credit pathway: credit never reaches the outcome, although the joint but-for test treats both features as a cause set.This production-based diagnosis is recovered by witness-based actual causality rather than by single-feature or joint but-for clauses.Source: §2.1 · p. 10
• Witnessed actual causality fixes mediators at their factual values, making Alice’s gender intervention change denial to approval while credit remains non-causal.The modified Halpern–Pearl definition combines factivity, witnessed necessity, and minimality over explicit suspect and witness pools.Source: §2.1 · pp. 10–11
• Actual causality remains computationally expensive because it searches over cause sets, alternative values, and witness sets, and its standard construction is deterministic.Placing a distribution over exogenous noise yields a probabilistic envelope around the same binary kernel without removing its computational demands.Source: §2.1 · p. 11

2.2 Probability of Necessity and Sufficiency

• PN, PS, and PNS extend causal attribution to stochastic models and distinguish necessity, sufficiency, and their conjunction, but they lack witness-based context sensitivity.The stochastic OBCB model includes probabilistic checking, skipped checks, and overridden loan decisions.Source: §2.2 · pp. 12–14
• The desired graded ordering gives active causes the greatest responsibility, preempted causes a positive but smaller residual share, and irrelevant variables the least.For Alice, gender should outrank credit; for Bob, credit should outrank gender, while gender remains nonzero because it opens the probabilistic checking pathway.Source: §2.2 · p. 13
• PNS ranks Alice’s credit above gender, 0.18 versus 0.045, despite the bank never evaluating her credit because check-failed blocked that causal path.This demonstrates that counterfactual interventions without fixing the mediator cannot register preemption.Source: §2.2 · p. 15, Table 2 · p. 15
• Conditioning mediators in PNS fails because counterfactual interventions override the conditioned context, motivating intervention-based context sensitivity through the witness mechanism.For Alice, PNS c(credit = bad, loan = F | gender = female) = 0.18, but the intervention changes check-failed and undoes the factual context.Source: §2.2 · p. 15
• ATE and CATE inherit PNS’s propagation problem and can assign identical effects to cases where the variable played different causal roles.On the stochastic OBCB, population ATE ranks credit (0.518) above gender (0.383), while Alice’s individual CATE values are 0.18 for credit and 0.045 for gender; CATE(gender | credit=bad) = 0.045 is identical for Alice and Bob.Source: §2.2 · pp. 15–16

2.3 Path Forward

• PCI combines actual causality’s witness mechanism with PN/PS/PNS’s probabilistic vocabulary and integrates over witness distributions instead of enumerating witness sets.This supports estimation from samples in probabilistic models with large witness pools and continuous variables.Source: §2.3 · p. 16
• PCI reverses the PNS ranking for Alice, recovering the intuitive attribution that favors gender over credit.The section states that the complete example and computation are developed in Section 3 and Appendix A.Source: §2.3 · p. 16
• By weighting witness sets by probability, PCI replaces actual causality’s binary existential verdict with graded, context-sensitive responsibility that downweights causally inert variables.For Alice, witnesses that pin check-failed favor gender, while almost none favor credit.Source: §2.3 · p. 16

3 Causal Impact: Definitions

• PCI models individual counterfactual attribution with PSCMs whose structural assignments, exogenous distribution, acyclicity, and unique solutions make causal queries well-posed.Interventions replace structural equations while preserving exogenous noise, and fixed-noise interventional outcomes are represented by point masses.Source: §3 · pp. 17–18
• Suspect sets identify candidate causes, witness sets hold contextual variables fixed, and pruning witnesses rather than causes preserves the tested claim and recovers the unconditional but-for limit.This convention follows Halpern’s actual-cause definition [Halpern, 2016] and prevents conflicts when a variable appears in both sets.Source: §3 · pp. 18–19
• PCI evaluates context-sensitive necessity under alternative suspect values and sufficiency under factual suspect values, with witnesses fixed to isolate causal channels and score responsibility by outcome changes.The OBCB example uses gender and credit as suspects and check-failed as the witness to expose the causal path beyond the checking bottleneck.Source: §3 · p. 19
• The variable-selection distribution Γ samples suspect and witness subsets, making interaction order and Monte Carlo search explicit rather than summing over every coalition.Cardinality limits J and K can target singleton or higher-order interactions; unlike SHAP’s observational integrand, PCI uses structural-model interventions, so attribution follows causal paths [Shapley, 1953; Janzing et al., 2020].Source: §3 · pp. 19–21
• The alternative-value distribution Δ excludes the factual value and is designed to balance model plausibility, outcome-independence, and context-sensitivity, with ε-excision providing a continuous-valued construction.If the excised support is empty, Δ is undefined; implementations should detect this and shrink ε, signal an error, or use a non-degenerate fallback.Source: §3 · pp. 21–22
• ϵ-excision makes the required counterfactual departure explicit: for standard normal features, ϵ values 0.2, 0.5, and 0.8 exclude approximately 16%, 38%, and 58% of mass.The threshold parallels ROPE and may reflect effect size, measurement precision, or decision relevance; sensitivity analysis is advised when scale is uncertain.Source: §3 · p. 22
• PCI jointly measures sufficiency and necessity by restoring candidate causes to factual values versus substituting alternatives, using the same noise realization and witness configuration.The resulting sub-probability measure weights suspect-containing configurations through Γ and integrates over the probabilistic structural causal model.Source: §3 · pp. 22–23, Figure 4 · p. 25
• The causal impact score is the expectation of a measurable user-chosen kernel over the joint counterfactual law, allowing binary indicators, L1 distance, L2 distance, or other scores.When variables are binary and the suspect set contains only Xk, the PNS score aligns exactly with Pearl’s probability of necessity and sufficiency.Source: §3 · pp. 24, 26
• Operationally, PCI uses a Monte Carlo loop that samples noise, one shared Γ-configuration, and an alternative value, runs both counterfactual worlds, scores them, and averages.The estimator rescales the conditional mean by PrΓ[Xk ∈C] to preserve the score’s sub-probability normalization.Source: §3 · pp. 24–25
• PCI with the check-failed witness ranks Alice’s gender above credit (0.263 vs. 0.199), correcting PNS and witness-free PCI’s reversed rankings.The witness blocks the credit-check bottleneck when Alice was checked, isolating gender’s structural role.Source: §3 · p. 27, Table 3 · p. 27
• For Bob, witness-based PCI assigns gender 0.019 while PNS assigns 0.000, capturing gender’s indirect role in opening the credit evaluation.Credit remains ranked above gender under all methods, and the cross-individual comparison 0.263 > 0.019 holds for PCI.Source: §3 · p. 27

3.1 Continuous Outcomes and the General Causal Impact Function

• For continuous outcomes, PCI replaces binary indicators with distance-based absolute-difference scores, including necessity-only and sufficiency-only components whose sum gives the full score.The causal impact function is user-defined: experiments use necessity-only attribution for direct comparison with actual causality, while other analyses use both components.Source: §3.1 · p. 28
• PCI recovers PN, PS, and PNS in the binary single-suspect case, connects to Halpern’s actual causality, and supports separate necessity–sufficiency choices.Witnesses can break symmetry in overdetermination and undercutting, while the probabilistic components compose with Monte Carlo estimation.Source: §3.1 · p. 28

3.2 Back to the Running Example: Verifying the Desiderata

• Table 4 shows that PCI with witnesses satisfies all seven desiderata, whereas PCI without witnesses fails D-A-rank and PNS fails D-A-rank and D-B1.The check-failed witness distinguishes gender’s role in blocking evaluation from credit’s conditional role and prevents PNS from assigning Bob’s gender zero attribution.Source: §3.2 · pp. 28–29, Table 4 · p. 29
• PCI’s necessity and sufficiency marginals differ from Pearl’s PN and PS because they integrate over PU, average over suspect sets, and hold active witnesses fixed.Pearl’s general decomposition is PNS = P(x⋆, y⋆) PN + P(x′, y̸ = y⋆) PS; under exogeneity, PNS = PN · P(Yx⋆ = y⋆) = PS · P(Yx′ = y′), not generally PN · PS.Source: §3.2 · p. 29
• For Alice’s gender, PN = 0.045 versus PCI’s necessity marginal 0.394; for Bob’s gender, PN = 0 while PCI’s marginals are 0.030 and 0.637.The witness preserves the credit-check bottleneck and recovers gender’s indirect role in opening evaluation.Source: §3.2 · p. 29

4.1 PCI matches actual-causality verdicts on canonical archetypes

• PCI satisfies all ten canonical archetype desiderata, scoring active causes above preempted ones, keeping irrelevance at the noise floor, and reproducing Pearl’s within-scenario ranking.Figure 5 reports all ten dots inside their confidence bands, with pass threshold ϵ ≈0.151.Source: §4.1 · p. 30, Figure 5 · p. 30

4.2 SHAP and Causal SHAP

• PCI separates direct from indirect causal contributions, whereas SHAP and Causal SHAP can attribute backward and collapse both pathways into one number.Causal SHAP’s interventional correction repairs only D-YX; Table 5 summarizes the signal-with-mediation comparison.Source: §4.2 · p. 30, Table 5 · p. 31

4.3 The Differential Causal Effect

• PCI ranks age above application time throughout the credit-limit grid, while DCE usually reverses that ranking and changes with the time unit.DCE favors application time except in a narrow band near the sigmoid inflection point, where its gradient spikes.Source: §4.3 · p. 31, Figure 6 · p. 31

4.4 Scaling past exact actual causality

• Exact subset enumeration times out near 17 variables, whereas PCI scales to roughly 73–145 variables and maintains correct attribution above 0.85 through about 60 variables.PCI visits several orders of magnitude fewer cause–witness configurations under a 60-minute per-size cap.Source: §4.4 · p. 31, Figure 7 · p. 32

4.5 Continuous, dynamical outcomes

• Lockdown receives roughly twice masking’s responsibility in the continuous SIR outcome, leading in 18 of 20 worlds with a mean gap of +0.701.The standard error is 0.111, and the gap is driven by the necessity term under witness pinning.Source: §4.5 · pp. 31–32, Figure 8 · p. 32

4.6 A real-world deployed model

• On a deployed valuation model, PCI distributes structured context-sensitive attribution across internal variables, while SHAP concentrates responsibility on a few downstream variables.Because the model lacks independent ground truth, this case demonstrates feasibility rather than correctness.Source: §4.6 · p. 32, Figure 9 · p. 33
• Across settings, PCI matches canonical actual-causality verdicts and extends beyond them to tractable graded explanations for continuous and deployed models.The full comparative protocols and derivations appear in Sections 5–8 and Appendices A–H.Source: §4.6 · p. 32

5 Examples and Comparison to SHAP and Causal SHAP

• PCI differs from SHAP through structural counterfactual evaluation and witness mechanisms, while plain SHAP’s observational conditioning can attribute correlated non-causes and both methods explain deviations from population baselines rather than factual outcomes.Causal SHAP applies do-calculus corrections for downstream dropped features, but the section identifies remaining gaps in causal direction and individual-outcome responsibility.Source: §5 · pp. 33–34

5.1 OBCB: SHAP and Causal SHAP

• In OBCB, SHAP assigns Alice’s credit greater responsibility than gender, |0.174| > |0.107|, because observational marginalization inflates credit’s attribution through the check-failed mechanism.PCI uses the factual check-failed value to isolate gender’s upstream causal role and recover the correct ranking; Causal SHAP coincides with SHAP when only exogenous roots are features.Source: §5.1 · pp. 35–36
• When check-failed is added as an OBCB feature, plain SHAP becomes undefined for both individuals, while Causal SHAP remains finite for Alice but gives |ϕ_credit| = 0.176 > |ϕ_gender| = 0.097 and remains undefined for Bob.These failures arise because the feature set contains the structurally impossible combination good credit and failed check, which plain SHAP samples independently and interventions can also create.Source: §5.1 · p. 37

5.2 Signal with Mediation

• The signal-with-mediation example models X → M → Y with independent Gaussian noise and distinguishes the direct role of M from X’s indirect path through M.The adopted desideratum requires |R(M ⇝Y)| > |R(X ⇝Y)|, while emphasizing that separating the roles matters more than the chosen inequality direction.Source: §5.2 · p. 38

5.3 Computations and Desiderata Analysis

• The two mediation instances contrast nontrivial predictions at X⋆ = M⋆ = Y⋆ = 1 with a baseline-collapse case at X⋆ = M⋆ = Y⋆ = 0.5, where SHAP’s attribution budget is zero.All three methods admit closed-form evaluation for Instance 1, and the resulting attributions populate Table 9.Source: §5.3 · p. 39
• In the mediation computations, plain SHAP assigns ϕ_Y = 0.139 > 0 when explaining target X, even though Y does not cause X, because observational information is symmetric under causal reversal.Observing Y = 1 predicts X = 1 through the chain, but SHAP cannot distinguish informativeness from causation.Source: §5.3 · p. 39
• SHAP and Causal SHAP correctly attribute positive values to direct or upstream causes, but Causal SHAP fixes D-YX while failing D-YM and reallocating spurious attribution in D-MX.Causal SHAP cannot remove observationally predictive inputs already used by the prediction model; in D-MX, attribution shifts from X to non-causal M.Source: §5.3 · pp. 39–40
• PCI satisfies all six single-variable desiderata without witnesses and ranks M →Y above X →Y, though the 0.034 margin is structurally fragile.The ranking gap comes entirely from sufficiency-world noise asymmetry; perturbing noise variances or the contrast function could reverse it.Source: §5.3 · p. 40
• With witnesses, PCI widens the M →Y versus X →Y gap from 0.034 to 0.133, making the ordering robust but dependent on different witness sets.The reported values are PCI(X →Y | W={M}) ≈0.186 and PCI(M →Y | W={X}) ≈0.319.Source: §5.3 · p. 40
• At the baseline, SHAP collapses every attribution to zero because its prediction-minus-mean budget vanishes, whereas PCI preserves positive causal and zero non-causal verdicts.PCI directly tests whether varying a suspect moves the realized outcome, independent of whether the factual prediction equals its population mean.Source: §5.3 · pp. 40–41, Table 10 · p. 41

5.4 Summary Across Both Examples

• Across both examples, SHAP variants fail because conditioning cannot exclude non-causal model features and Shapley averaging cannot distinguish direct from indirect paths; PCI adds witnesses, suspect-set sampling, and realized-outcome scoring.PCI’s sufficiency-world path-noise asymmetry also creates the D-MXY structural gap, which appears without witnesses and widens under a witness rule.Source: §5.4 · p. 41

Formal correspondence

• Under regular Γ, full-support ∆, and stated continuous-case conditions, PCI assigns positive kernel mass to every Halpern actual cause and recovers factivity and context-sensitive necessity from positive mass.The correspondence excludes deterministic-function redundancies in suspect sets and requires pointwise access to the full PSCM and counterfactual outcomes.Source: §6 · pp. 42–45

Scope and limitation

• The correspondence is necessity-only: PCI’s sufficiency factor acts as a factivity check, while its graded witness-conditioned construction supplies the substantive sufficiency component absent from Halpern’s definition.This component is intended to align with Beckers’ NESS program rather than extend Halpern’s actual-causality notion.Source: §6 · p. 42

Minimality correspondence

• Minimality is recovered when witness weights are uniform, suspect-set weights strictly decrease with cardinality, and necessity dilution ensures larger candidate sets cannot gain necessity mass.Under these conditions, both joint and witness-marginalized PCI optimizers are actual causes, via Theorem 29 and Corollary 30.Source: §6 · pp. 45–46
• Necessity dilution holds automatically in single-pivot, suitably factored, and conjunctive models but fails under disjunctive overdetermination, where only the full candidate set may flip the outcome.In disjunctive overdetermination, no proper subset is an actual cause, so Proposition 32 rather than the minimality theorem is required.Source: §6 · p. 47

Integrated attribution

• Integrated PCI maximization need not identify every context-specific actual cause: an actual cause in one noise setting can be absent from arg max_C P(C⇝φ).For Y = A ∨B with uniformly distributed binary parents, {A,B} is an actual cause when A = B = 1, but its bit-flip alternative changes the outcome only in that setting and not in the other outcome-producing settings.Source: §6 · p. 47

Counterexample

• Integrated PCI scoring can exclude an actual cause because Γ penalizes larger cardinality-minimal sets, whereas actual causality requires subset-minimality.In the example, C† remains an actual cause at u11 but is excluded from Ξmax; the conclusion is robust to an alternative Δ, though proportionality constants change.Source: §6 · pp. 47–48

Recovery procedure

• Dropping maximization, enumerating positive-Φ candidates, and filtering for factivity and subset-minimality recovers exactly the actual causes contained in S.Proposition 32 establishes this procedure; positive Φ candidates satisfy context-sensitive necessity, while the filter enforces the remaining actual-causality conditions.Source: §6 · p. 48

Guarantees

• Under explicit hypotheses on S, Γ, and ∆, the guarantees include the forward direction: an actual cause implies Φ > 0.This forward direction is identified as Theorem 26.Source: §6 · p. 48

Empirical correspondence

• The section’s empirical correspondence benchmark compares the necessity-only PCI estimator with exact subset enumeration on a scaled stone-throwing problem in Appendix F.This benchmark is identified as the empirical counterpart of the theoretical correspondence, without reported results here.Source: §6 · p. 48

7 Relation with the Probability of Actual Causality

• Pearl’s probability of actual causation averages a binary AC predicate over posterior noise states, while PCI averages a continuous necessity–sufficiency kernel over broader causal-design distributions.PCI’s four aggregation dimensions are suspect sets, witness sets, alternative values, and noise; Pearl’s definition aggregates only over posterior noise for a fixed cause and outcome.Source: §7 · p. 49, §7.5 · p. 54

7.1 Pearl’s Worked Example: The Desert Traveller

• Pearl’s desert-traveller analysis assigns the actual cause according to the forensic noise state: poisoner when cyanide killed and shooter when dehydration killed.The corresponding posterior probabilities of actual causation saturate at 1 for the forensic operative cause.Source: §7.1 · pp. 49–50

7.2 PCI on the Basic Desert Traveller

• PCI matches Pearl’s cause ranking in each basic desert-traveller scenario but adds graded magnitudes, assigning the dormant cause 5/24 ≈0.21 rather than zero.Pearl’s P(caused) saturates at 0 or 1, whereas PCI’s J never reaches 1 here because witness-subset averaging dilutes each cause’s score.Source: §7.2 · p. 51

7.3 PCI Expectation vs the AC Machinery

• PCI replaces AC’s existential witness search with a Γ-weighted expectation over candidate causes, witnesses, alternative values, and noise, enabling Monte Carlo estimation with variance independent of witness-set size.AC may require checking up to 2^n witness subsets per noise state, whereas PCI estimates its expectation at a user-chosen computational budget.Source: §7.3 · pp. 51–52

7.4 Weak-Poison Variant: Path-Reliability and Cross-Scenario Reliability

• PCI captures all three weak-poison responsibility intuitions, including reliability sensitivity and cross-scenario ranking, while Pearl’s binary score saturates at 1 for each operative cause.PCI ranks the shooter in Scenario B at 1/2 above the poisoner in Scenario A at 5/24; the poisoner falls from 1/3 to 5/24 when cyanide becomes stochastic.Source: §7.4 · pp. 52–54, Figure 13 · p. 53

7.5 Further Differences Between Pearl’s Probability of Causation and PCI

• PCI directly generalizes Pearl’s PN and PS operations through necessity and sufficiency factors, rather than probabilifying the AC predicate used by Pearl’s Definition 10.3.5.The two constructions therefore inherit different parts of Pearl’s toolkit: PCI is a context-sensitive, multi-variable PN/PS extension, whereas Pearl’s definition is a probabilistic wrapper around AC.Source: §7.5 · p. 54

7.6 Other probabilistic extensions of actual causation

• Compared with Fenton-Glynn [2017] and Fang [2022], PCI adds sufficiency, distributes contingencies and alternative values, and uses a continuous kernel rather than a thresholded necessity-only indicator.Those neighboring accounts probabilify only witnessed necessity, whereas PCI treats their necessity test as one regime within a broader PN/PS-style measure.Source: §7.6 · p. 55
• FANS and LEWIS use Pearlian probabilities for model explanations or population-level recourse, while PCI applies its probability pair to causal responsibility for a realized outcome in a structural model.Watson et al. [2021] similarly adapt PN/PS for black-box classifier explanations but do not expose PCI’s Γ-sampled witness-set dimension.Source: §7.6 · pp. 55–56
• Kawakami et al. [2024] extend PNS, PN, and PS to continuous and vector-valued domains, whereas PCI obtains continuous necessity–sufficiency scores by simulating both counterfactual worlds in a fitted model.Kawakami et al. identify population-level quantities from observational distributions under exogeneity and monotonicity; PCI instead targets the single-case reading and shifts the burden to model correctness.Source: §7.6 · pp. 55–56
• PCI differs from FANS and LEWIS by jointly retaining necessity and sufficiency, averaging across candidate cause sets, holding witnesses fixed, and averaging alternatives through Δ rather than optimizing or using one baseline.These choices target token-level, context-sensitive attribution for an individual realized outcome when a full causal model is available.Source: §7.6 · p. 56
• PCI and Beckers [2025] represent uncertainty differently: PCI averages probabilistic exogenous noise, whereas nondeterministic models use “must” and “might” modalities over admissible mechanism resolutions.The paper suggests, without claiming a formal correspondence, that full-support probabilities could map “might” to positive-probability change and “must” to probability-one change, with PCI supplying graded magnitudes.Source: §7.6 · pp. 56–57

7.7 Actual-causality-adjacent quantities

• PCI conjectures that many actual-causality-adjacent quantities are reparametrizations of its knobs Γ, T, Δ, ci, and PU rather than independent primitives.The proposed correspondences include responsibility and blame, causal explanation, sufficient and counterfactual explanations, harm, and causal sufficiency, with some notions requiring natural generalizations such as safeguarded variables.Source: §7.7 · pp. 57–58
• PCI preserves the modelling commitments of neighboring measures by relocating conventions such as responsibility weights, binary thresholds, non-domination, and stipulated defaults into explicit downstream choices.The paper presents this unification as a conjecture rather than a completed account and leaves a full treatment to future work.Source: §7.7 · pp. 57–58

Contributions and validation

• PCI recovers exact actual-causality verdicts under matching Γ and ∆ while remaining estimable by sampling beyond tractable AC enumeration.The support of P_TΦ matches factivity and context-sensitive necessity, and subset-minimality recovers AC verdicts exactly; Appendix F confirms this at intractable enumeration scales.Source: §8 · p. 58

Framework generalization

• PCI generalizes necessity and sufficiency by sampling suspect sets, witness sets, alternative values, and customizable causal-impact functions, supporting continuous variables and graded responsibility.Its binary instantiation recovers PN and PS, while separate components expose different causal archetypes on a continuous synthetic model.Source: §8 · p. 58

Limitations and scalability

• PCI requires a trained probabilistic causal model, inherits its misspecification, and faces Monte Carlo variance that can approach exhaustive enumeration when revealing witness sets are rare.Inert witnesses create larger revealing neighborhoods, but whether this generally rescues sampling remains open; exposing Γ, ∆, and ci also places a burden on practitioners.Source: §8 · p. 59

Practical specification

• The recommended default is to place live candidate causes in S and all remaining endogenous non-outcome variables in W, moving suspected mediators into S and restricting W only when cost requires it.Variables downstream of S and upstream of Y are the next-cheapest defensible witness restriction because variables off every S →Y path are causally inert.Source: §8 · pp. 59–60

Comparisons with attribution methods

• PCI’s witness-based interventions avoid SHAP and Causal SHAP failures on overdetermination and continuous mediation, with synthetic ground truth and a production model showing the same qualitative pattern.The production AVM case study is a feasibility witness rather than quantitative evidence; the apples-to-apples comparison is based on Appendix D’s synthetic benchmark.Source: §8 · p. 60

A Computations for the OBCB Running Example

• The appendix provides worked computations for the stochastic OBCB model, including Alice’s individual PN and PS values and the closed-form SHAP value function.The notebook obcb_computations reproduces all values, while remaining Table 2 entries are obtained analogously.Source: A Computations for the OBCB Running… · p. 64

A.1 Alice’s probability of necessity

• Alice’s factual state is female gender, bad credit, and loan F; the counterfactual sets gender to male while keeping credit bad.Source: §A.1 · p. 64
• Under the intervention, the model checks men with probability 0.9.This is the first branch of Alice’s counterfactual computation.Source: §A.1 · p. 64
• When Alice is checked, bad credit makes check-failed = 1, and the male loan-probability table assigns loan-if-checked = 1 probability 0.05.Source: §A.1 · p. 64
• When Alice is not checked, loan equals 0 deterministically because loan = loan-if-checked · check.Source: §A.1 · p. 64
• Alice’s probability-of-necessity computation combines the checked and unchecked counterfactual branches.Source: §A.1 · p. 64

A.2 Alice’s probability of sufficiency

• The individual-level PS computation fixes Alice’s credit as bad, conditions on male gender and a granted loan, then intervenes to restore female gender.The counterfactual question is whether the loan is denied again after restoring the factual gender.Source: §A.2 · p. 64
• Conditioning on a granted loan restricts the shared check-noise variable to the region producing a check for men, making the restored female check probability 2/9 rather than 0.2.This altered branch probability does not affect the result because both checked and unchecked branches deny the loan.Source: §A.2 · pp. 64–65
• The resulting probability of sufficiency is 1: restoring female gender guarantees loan denial under the conditioned counterfactual.If checked, the relevant loan probability is 0; if not checked, the loan is deterministically denied.Source: §A.2 · p. 65

A.3 SHAP value function for Option A

• Option A’s SHAP value function averages model outputs over coalitions, and the supplied feature values produce coalition values and rejection-attribution magnitudes via the closed-form Shapley formula.The four model values are f(F, bad) = 0.000, f(F, good) = 0.180, f(M, bad) = 0.045, and f(M, good) = 0.900.Source: §A.3 · p. 65

B SHAP and Causal SHAP: Definitions and Signal-with-Mediation Computations

• The appendix derives distributional facts, conditional expectations, and per-target SHAP and per-pair PCI computations for the Gaussian chain X → M → Y.The factual instance is Instance 1, X⋆ = M⋆ = Y⋆ = 1.Source: B SHAP and Causal SHAP: Definitions… · p. 65

B.1 SHAP and Causal SHAP

• SHAP assigns each feature its average marginal contribution across uniformly random orderings while satisfying efficiency, symmetry, dummy, and linearity axioms.Lundberg and Lee [2017] apply this framework to model predictions using expected outputs conditional on features held at their factual values.Source: §B.1 · pp. 65–66
• Plain SHAP conditions dropped features on retained features, whereas the marginal-independent alternative averages each dropped feature separately; the two agree only under independence, and the choice is contested [Janzing et al. 2020].Option B explicitly uses the marginal-independent form to expose undefined values on structurally impossible feature combinations.Source: §B.1 · p. 66
• Causal SHAP replaces conditional averaging with interventional expectations, preserving marginal distributions for non-descendants and intervening on descendants while retaining all four Shapley axioms [Heskes et al., 2020].Its Shapley formula is otherwise unchanged.Source: §B.1 · p. 66

B.2 Distributional facts

• For the Gaussian chain, Var(M) = 0.35, Var(Y) = 0.45, Cov(X, M) = 0.25, Cov(X, Y) = 0.25, and Cov(M, Y) = 0.35.Source: §B.2 · p. 66

B.3 Conditional expectations

• The chain’s conditional expectations include E[Y | X, M] = M, E[M | X, Y ] = 0.5X + 0.5Y , and E[X | M, Y ] = E[X | M] by d-separation.Source: §B.3 · p. 66

B.4 Plain SHAP

• Plain SHAP then computes coalition values and applies the two-feature Shapley formula separately for targets Y, M, and X using the derived conditional expectations.For target M, the supplied values include v(∅) = 0.5, v({X}) = 1, v({Y }) = 0.778, and v({M, Y }) = 0.857.Source: §B.4 · p. 67

Target Y

• For target Y, Causal SHAP exactly matches plain SHAP, yielding ϕX = ϕM = 0.25.All coalition values are identical under both methods: vcausal(∅) = 0.5, singleton values equal 1, and the joint value equals 1.Source: §B.5 · p. 67

Target M

• For target M, intervening on Y gives vcausal({Y}) = 0.75, differing from plain SHAP’s 0.889 because Y does not cause X.The intervention leaves P(X) unchanged, while the X coalition agrees with plain SHAP because Y is downstream of X.Source: §B.5 · p. 67

Target X

• For target X, intervening on Y gives vcausal({Y}) = 0.5, differing from plain SHAP’s 0.778 because M is not a descendant of Y.The M coalition matches plain SHAP at 0.857, and the joint coalition value is also 0.857.Source: §B.5 · p. 67

B.6 PCI

• PCI evaluates necessity and sufficiency worlds with fresh exogenous noise, replacing suspects in one world and restoring them to factual values in the other.For target B, the candidate set is the two input features, with remaining inputs treated as witnesses; subset distributions are uniform over non-empty candidate subsets and witness subsets.Source: §B.6 · p. 67
• For R(X ⇝Y), three accepted intervention pairs affect X with weight 1/4, while the witness pair fixing M at m⋆ equalizes both worlds and contributes zero.The remaining accepted pairs allow M to propagate freely, and expectations are computed using the folded-normal formula.Source: §B.6 · p. 68
• The supplied calculation reports 4(0.320 + 0 + 0.425) ≈ 0.186.Source: §B.6 · p. 68
• Because X is exogenous, intervening on M or Y cannot alter its structural equation, so R(Y ⇝X | W={M}) = 0.Necessity and sufficiency distributions therefore coincide.Source: §B.6 · p. 68

C Setup for the Desert-Traveller PCI Computation

• The desert-traveller appendix runs PCI Monte Carlo on scenario-specific models, and hand-rolled enumeration matches Tables 12 and 14 within Monte Carlo noise.The comparison covers the basic variant and weak-poison variant using the component choices specified in the appendix.Source: C Setup for the Desert-Traveller PCI… · p. 68

C.1 Structural model

• Under Pearl-comparable forensic conditioning, the model pins u to each scenario’s realization and defines Boolean variables X, P, c, d, and Y.The structural equations are c = P(u′ ∨X′), d = X(u ∨P′), and y = c ∨d, with c and d representing cyanide and dehydration paths.Source: §C.1 · p. 68

C.2 PCI components

• PCI uses suspects {X, P}, mediators {c, d} as witnesses, cardinality-uniform witness subsets, deterministic 1 →0 alternatives, and forensic-posterior noise without conditioning on Y.The non-cause suspect follows the SCM under its own noise coin rather than being intervened on or hard-coded.Source: §C.2 · p. 68
• In the weak-poison variant, PCI adds vC to the witness pool and pins Scenario A’s posterior at u = 0, ξ = 1, while Scenario B pins u = 1.ξ is structurally irrelevant in Scenario B because the cyanide path is dormant.Source: §C.2 · p. 68

Model construction

• The synthetic model combines linear necessary-and-sufficient, overdetermined, preempted, and irrelevant branches, with independent roots feeding outcome E.The preempted branch is gated by |L2| ≤τ, while D is sampled but has no effect on E and supplies the noise floor.Source: D Synthetic Evaluation Across Linear,… · p. 69, Figure 14 · p. 70

Measurements

• Excess scores relative to irrelevant control D normalize regime-dependent co-intervention noise, allowing stable comparison of necessity and sufficiency across the two factual cases.The evaluation uses separate ∆N and ∆S components and bootstrap confidence intervals with correlations preserved across suspects and D.Source: D Synthetic Evaluation Across Linear,… · pp. 69–70

Factual cases

• Case 1 tests preemption and contestable overdetermination, whereas Case 2 tests gated activation and the necessity–sufficiency gap for a dominant overdetermined winner.Together, the cases assess whether P’s score follows its structural role: noise-floor behavior when gated off and positive impact when gated on.Source: D Synthetic Evaluation Across Linear,… · pp. 69–70

Results

• Ten of twelve desiderata pass at N = 20 000 Monte Carlo draws, with all ten forest-plot claims inside their pass bands against analytic ground truth.The threshold is ϵ ≈0.151, derived from bootstrap uncertainty; the tightest claim is ∆N(P) = +0.250 in Case 2, above threshold.Source: D Synthetic Evaluation Across Linear,… · pp. 70–71, Table 15 · p. 71

Diagnostics

• The cross-regime P flip is statistically unresolved, and P fails the quantitative Case 2 match to L1, so only their weaker shared-above-noise-floor behavior is supported.The flip is +0.139 with CI [−0.081, +0.365], while ∆N(P)−∆N(L1) is −0.267 with CI [−0.441, −0.106].Source: D Synthetic Evaluation Across Linear,… · p. 71

Method comparison

• PCI and DCE are not direct competitors: DCE measures a local rate of change, while PCI integrates necessity- and sufficiency-world densities to score responsibility.PCI can remain non-zero in flat regions because its score depends on the factual value’s position relative to those densities.Source: E Comparison to Differential Causal… · p. 72
• PCI requires only a sampleable causal model and is largely scale-invariant, whereas DCE requires differentiability and inherits the units of both inputs and outcomes.PCI’s scale invariance depends on specifying alternative values through probability-mass excision or standardized units rather than a fixed raw-distance threshold.Source: E Comparison to Differential Causal… · p. 72

Credit-limit assignment

• The credit-limit model combines a sigmoidal age effect with a small time-of-day sinusoidal modulation, motivating age as the expected dominant attribution.The experiment varies time units and age priors to separate reparameterization effects from genuine changes in factual unusualness.Source: E Comparison to Differential Causal… · p. 72, Figure 15 · p. 72
• Age outscores time of application by roughly 3/2 in every setting, hours and minutes produce essentially identical PCI scores, and the age-40 prior raises all scores.The higher scores under the age-40 prior reflect the greater unusualness of the factual age.Source: Table 16 · p. 73, E Comparison to Differential Causal… · p. 73

PCI vs. DCE across the input grid

• PCI consistently ranks age above time of application, while DCE favors time over most of the age-30 grid and changes substantially under hour-to-minute rescaling.DCE favors age only near the sigmoid’s inflection point, where its gradient spikes; minutes flatten the time DCE by roughly 60.Source: E Comparison to Differential Causal… · p. 73

Benchmark construction

• The benchmark replicates the canonical Sally/Billy overdetermination structure n times, yielding 4n + 1 variables with closed-form responsibility ground truth at each site.Overdetermination, undercutting, and witness-mediated symmetry-breaking persist at every site.Source: F Scaling Actual-Cause Computation: An… · p. 74

Approximate PCI estimator

• PCI samples suspect and witness interventions under shared noise, scores necessity-world outcome changes, and selects the higher-scoring rival at each site.The protocol varies sample budget, witness interventions, and cardinality bounds, including fixed bounds of 4 and dynamic bounds of 4n.Source: F Scaling Actual-Cause Computation: An… · pp. 74–75

Results

• 17 variables marks exact AC’s timeout, whereas approximate PCI reaches roughly 73–145 variables and evaluates several orders of magnitude fewer configurations.The approximate methods visit at most a few hundred unique configurations per problem size, with the gap widening as n grows.Source: F Scaling Actual-Cause Computation: An… · p. 75, Figure 17 · p. 76
• 0.941 versus 0.861 correct-attribution rate favors witness pinning at matched budget, with witness runs leading at 22 of 26 shared sizes.Witness-using methods remain above 0.85 through 73–89 variables, while witness-free runs start near 0.67 at n = 1.Source: F Scaling Actual-Cause Computation: An… · pp. 75–76

Scope of the conclusion

• The favorable scaling is qualified by independent binary sites, exactly one responsible rival per site, witness-set supersets, and closed-form ground truth; broader comparisons remain future work.The study defers SHAP and causal SHAP comparison on this scaled problem and calls for continuous or otherwise less structured benchmarks.Source: F Scaling Actual-Cause Computation: An… · p. 76

PCI thin-search procedure

• The benchmark applies PCI thin search to a Bayesian SIR model with continuous overshoot outcomes, using 2,500 sampled regimes per suspect and a necessity–sufficiency score.Classical actual causality offers no comparison here because the outcome is continuous; PCI instead estimates graded causal responsibility over interventional regimes.Source: G Dynamical SIR Benchmark: PCI on… · pp. 77, 78, 80

But-for analysis

• But-for probabilities are nearly indistinguishable for lockdown-only and mask-only regimes—0.84 versus 0.81—because their aggregate effects obscure asymmetric, context-dependent policy efficiencies.Joint intervention has probability 0.76 and no intervention 0.07; lockdown and mask efficiencies interact asymmetrically, with masking efficiency 0.45 alone versus 0.1 under lockdown.Source: G Dynamical SIR Benchmark: PCI on… · p. 77, Figure 18 · p. 78

Single-world results

• PCI breaks the but-for symmetry by ranking lockdown well above mask, because removing lockdown moves overshoot farther from factual than removing mask.The necessity means are 8.276 for lockdown and 7.536 for mask, accounting for 0.740 of the 0.835 total score gap, while sufficiency differs by only 0.095.Source: G Dynamical SIR Benchmark: PCI on… · pp. 78–79

Robustness across factual worlds

• Lockdown leads masking in 18 of 20 prior-drawn factual worlds, with mean PCI scores of +0.819 versus +0.118 and a mean gap of +0.701 (standard error 0.111).The factual worlds span β ∈[0.0160, 0.0398], γ ∈[0.481, 0.525], and overshoots from 9.86 to 33.80 people.Source: G Dynamical SIR Benchmark: PCI on… · p. 79, Figure 20 · p. 80

Effect of context

• Masking is context-sensitive: its PCI score is −0.161 when lockdown efficiency is fixed but 1.497 when that efficiency is free, whereas lockdown scores 0.494 and 1.684 in the corresponding contexts.Lockdown remains a cause in both regimes, while mask changes status depending on whether its partner-policy efficiency can adapt.Source: G Dynamical SIR Benchmark: PCI on… · pp. 79–80, Figure 21 · p. 80

Scope and limitations

• The study applies PCI to a deployed residential valuation model without ground truth or exact actual-causality enumeration, so it reports feasibility and qualitative divergence rather than quantitative accuracy.Quantitative validation remains confined to the controlled synthetic benchmark of Appendix D.Source: H Scaling PCI to a Real-World… · p. 81

Setting

• The illustrative New York prototype shares the full system’s architecture and causal-kernel construction, while the production-scale feasibility claim concerns the larger national AVM.The prototype uses a 25-node, 99-edge DAG, for which exact actual-causality enumeration is already computationally infeasible.Source: H Scaling PCI to a Real-World… · p. 81, Figure 22 · p. 82

Causal model

• The causal model combines manually expert-built DAG structure with learned categorical or continuous causal kernels and an SVGP outcome model.Kernel functions are deep neural networks, while categorical variables are embedded into the continuous input space for the sparse variational Gaussian Process.Source: H Scaling PCI to a Real-World… · p. 81

Feasibility at production scale

• Around 25,000 Monte Carlo samples suffice for convergence, and scoring 50 data points takes around 60 minutes on commodity hardware, establishing production-scale computational feasibility.Samples are drawn once and reused to compute attribution scores for any suspect, despite added latent dimensions in the expanded model.Source: H Scaling PCI to a Real-World… · p. 81

Qualitative comparison to SHAP

• SHAP concentrates attribution on deep, summary-statistic-like downstream nodes, whereas PCI distributes attribution more evenly across the upstream causal structure.Figure 9 compares distribution shapes only because feature identities are withheld and each panel ranks its own top variables independently.Source: H Scaling PCI to a Real-World… · p. 83