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A Computationally Feasible Framework for Causal Probabilistic Explanation

Rafal Urbaniak, Sam Witty, Daniel Waxman, Andy Zane, Poorva Garg, Emily Bunnapradist, Sankaran Vaidyanathan, Jack Feser, Drew Lehe, Eli Bingham

arXiv:2609.04177v1cs.AI

TL;DR

Existing attribution methods either make principled causal judgments only at toy scales or ignore enough causal structure to produce misleading explanations. PCI reframes explanation as Monte Carlo estimation over candidate causes, witness sets, and counterfactual values, yielding graded causal scores. Across formal and empirical comparisons, it recovers actual-causality judgments and avoids characteristic SHAP-family ranking failures, while remaining subject to posterior, sampling, and score-comparability limitations.

  • Problem

    Existing methods trade off causal context sensitivity against scalability, and predictive attribution can conflict with the causal pathways active for a specific outcome.

  • Method

    PCI averages counterfactual causal impact over distributions of candidate causes, witness sets, and alternative values in probabilistic causal models.

  • Results

    PCI recovers actual-causality verdicts under specified conditions and avoids characteristic SHAP-family failures in synthetic, continuous-mediation, and production-grade comparisons.

  • Takeaways & Limitations

    PCI provides a tractable, causally grounded framework for graded explanations that connects actual causality with probability-based attribution.

  • Takeaways & Limitations

    PCI depends on the quality of the model’s posterior, finite sampling can miss rare witness sets, and asymmetric Γ can inflate raw scores through inclusion frequency alone.

Abstract

from arXiv · show

Explaining why a specific outcome occurred, and which inputs deserve the blame or credit, is central to philosophical, scientific, and policy analysis. Existing tools split into two camps. The theory of actual causality (AC) gives principled verdicts, but only for toy-sized models, because computing them requires enumerating counterfactual scenarios. Scalable attribution methods like SHAP (or even causal SHAP) at least partially ignore the causal structure that generated the data, and can give answers that conflict with a careful causal analysis. We close this gap with Probabilistic Causal Impact (PCI). PCI builds on actual causality and on Pearl's notions of probability of necessity and sufficiency, but recasts the question of explainability as an estimation problem on a probabilistic causal model that is easily approximated via Monte Carlo. By specifying a distribution over "candidate explanations," a distribution over counterfactual values, and a scoring function, PCI provides tractable, causally grounded, graded explanations, generalizing AC and Pearl's probability of causation as degenerate cases. We evaluate PCI in synthetic and real-world examples, spanning consistency checks with AC, scaling experiments, complex continuous-valued dynamical systems, and a real-world deployed causal machine learning model trained on millions of datapoints.

1 Introduction

Existing attribution methods either provide principled but computationally intractable causal verdicts or scale while confusing predictive importance with causal responsibility. PCI addresses this gap with graded, context-sensitive causal scores estimated by sampling, while retaining formal connections to actual causality and probability-of-causation methods.

  • Motivation: SHAP and related scalable tools can mistake predictive correlation for causal responsibility, producing causally uninterpretable attributions.In the Old Boys’ Club Bank example, they split responsibility between gender and credit even though the bank never checked Alice’s credit.
  • Motivation: Actual causality can identify context-specific causes, but exhaustive witness-set search makes it computationally intractable beyond toy examples.Its all-or-nothing formulation also lacks a graded notion of causal explanation.
  • PCI: PCI combines actual causality’s context sensitivity with probability-of-causation methods by replacing existential witness-set search with an expectation over weighted candidate-cause and witness distributions.Witnesses hold selected variables at their factual values while PCI varies a candidate cause, preserving active-path information while enabling Monte Carlo estimation.
  • PCI: PCI’s witness mechanism ranks active causes above correlated or downstream variables and distinguishes reliable causes from unreliable ones across forensic scenarios.Without witnesses, PCI can reproduce SHAP’s ranking error; with them, gender outranks credit in Alice’s case and causal responsibility can vary across scenarios.
  • Limitations: Monte Carlo sampling introduces an explicit finite-budget risk: rare witness dependencies may be missed, and in the worst case sampling can be as costly as enumeration.The sample budget and witness-cardinality bound control this tradeoff, and increasing either reduces the chance of missing rare witness sets.
  • Formal and empirical scope: Under specified choices of Γ and ∆, a positive PCI score is equivalent to actual causation in Halpern’s sense, while the framework also captures graded responsibility patterns that binary indicators cannot express.The paper positions PCI as recovering formal actual-causality verdicts while extending probability-based explanations to graded magnitudes.

2 Causal Impact: Motivations

Causal attribution must distinguish general causal strength from whether a pathway was active in a particular case. PCI addresses this by combining intervention-based context sensitivity with probabilistic, graded estimation that avoids exhaustive witness-set search.

  • Motivation: Type-level causal effects can miss token-level responsibility when a generally causal feature was inactive in the specific outcome.The Alice example motivates distinguishing whether a cause tends to affect an outcome from whether its contribution was live in the case being explained.
  • Motivation: Actual causality identifies active pathways using witness variables, but its binary verdicts and exhaustive search limit scalability beyond deterministic toy models.Witness sets hold selected variables at their factual values; finding a suitable set can require searching exponentially many candidates.
  • Motivation: Pearl’s PN, PS, and PNS extend causal reasoning probabilistically but can misattribute Alice’s denial because interventions recompute the mediating check-failed variable.For Alice, PNS ranks credit above gender despite the bank never evaluating her credit, and conditioning does not preserve the relevant mediation context.
  • Motivation: ATE and CATE share this context-insensitivity: CATE(gender | credit=bad) = 0.045 for both Alice and Bob despite their different active causal pathways.The same value obscures that gender directly produced Alice’s rejection while merely opening Bob’s credit evaluation.
  • PCI: PCI combines witness-based intervention, PN/PS/PNS probabilistic counterfactuals, and distributed estimation to preserve pathway context without combinatorial enumeration.It integrates over witness sets rather than asking whether one validating witness exists, replacing exhaustive search with statistical estimation.
  • PCI: PCI recovers the intuitive Alice verdict while retaining nonzero credit attribution when women can still be checked probabilistically and distinguishing Alice’s and Bob’s causal roles.The framework assigns gender greater responsibility for Alice than for Bob, while recognizing credit as operative when Alice’s check occurs.

3 Causal Impact: Definitions

PCI defines causal impact by averaging a user-chosen score over necessity and sufficiency counterfactuals, sampled across candidate causes, witnesses, alternatives, and exogenous noise. Witnesses preserve the relevant causal context, while the framework generalizes Pearl’s probabilities and recovers actual-causality judgments.

  • Witness mechanism: Witness pruning removes overlaps with candidate causes without changing the causal claim under test, preserving the actual-causality constraint T ∩ C = ∅.Pruning witnesses leaves the candidate set intact and recovers the unconditional but-for baseline when every variable is a suspect.
  • Counterfactual construction: PCI evaluates necessity and sufficiency worlds for candidate causes while holding sampled witnesses fixed at their factual values.The necessity world uses alternative cause values; the sufficiency world uses factual values.
  • Counterfactual alternatives: Alternative-value distributions exclude the factual value and should be plausible under the model, outcome-independent, and sensitive to the factual context.A degenerate posterior may make such a distribution undefined, requiring explicit handling or a non-degenerate fallback.
  • Scoring: The impact kernel ci converts the joint necessity–sufficiency measure into a scalar whose expectation is the PCI score.Different kernels support necessity-only, sufficiency-only, or joint causal questions.
  • Running example: PCI with witnesses reverses the OBCB misranking of Alice’s gender and credit, while capturing Bob’s indirect gender contribution that PNS assigns zero.For Alice, witnessed PCI ranks gender above credit; for Bob, it returns a positive gender attribution.
  • Connections: PCI recovers Pearl’s PN, PS, and PNS and can recover Halpern’s actual-causality judgments, while witnesses distinguish overdetermined and undercutting causes.These are obtained as special cases through choices of candidate distributions, alternatives, witnesses, and impact kernel.

4 Empirical and Comparative Evaluation: Overview

PCI is evaluated against actual causality, SHAP-based attribution, DCE, and scaling limits across synthetic, dynamical, and deployed-model settings. It matches canonical causal verdicts, remains tractable beyond exact enumeration, and produces context-sensitive attributions in a large automated valuation model, although that deployment lacks independent ground truth.

  • Canonical archetypes: PCI recovers all ten tested actual-causality desiderata on synthetic linear, overdetermined, preempted, and irrelevance-control archetypes.Each dot falls inside its corresponding pass region at threshold ϵ ≈0.151.
  • SHAP comparison: PCI separates direct and indirect causal contributions that plain and Causal SHAP collapse into one attribution.Causal SHAP’s interventional correction repairs only part of the mismatch.
  • DCE comparison: PCI and DCE can disagree on sign and magnitude when local gradients misrepresent the counterfactual contrast in a non-monotone structural response.PCI’s realized-outcome reference keeps attribution aligned with the feature’s structural role in the evaluated contrast.
  • Scaling: Exact subset enumeration times out near 17 variables, whereas the PCI estimator continues to roughly 73–145 variables and maintains correct-attribution rates above 0.85 to about 60 variables.The scaling benchmark uses a 60-minute per-size compute cap and an undemanding sample budget.
  • Dynamical system: PCI assigns lockdown roughly twice the responsibility of masking for peak overshoot in a Bayesian SIR model, with lockdown ahead in 18 of 20 factual worlds.The mean responsibility gap is +0.701 with standard error 0.111.
  • Deployed model: In a deployed AVM trained on millions of points, PCI gives structured context-sensitive attributions while SHAP concentrates responsibility on a few downstream variables.Because the AVM has no independently known ground truth, this case demonstrates feasibility rather than correctness.

5 Examples and Comparison to SHAP and Causal SHAP

The examples show that PCI preserves causal, instance-specific responsibility where SHAP and Causal SHAP can confuse correlation, indirect paths, or impossible configurations with causation. Across OBCB and signal mediation, PCI’s witness mechanism and factual-outcome reference address these structural failures.

  • Comparison framework: PCI compares against SHAP and Causal SHAP on OBCB and continuous mediation examples, tracing their failures to structural gaps that PCI’s witness mechanism, suspect-set distribution, and realised-outcome reference address.The comparison tests explicit desiderata across both examples.
  • OBCB: SHAP ranks Alice’s credit above gender (0.174 > 0.107), because observational marginalization inflates credit attribution through the check-failed mechanism.PCI includes the factual check-failed value and recovers the correct ranking.
  • OBCB: Causal SHAP coincides with plain SHAP when OBCB features are exogenous, so both methods retain the same attribution failures under Option A.Under Option A, interventional and observational coalition values are identical.
  • OBCB: Adding the OBCB mediator does not recover the desideratum: Causal SHAP assigns Alice |ϕcredit| = 0.176 above |ϕgender| = 0.097 and is undefined for Bob.The undefined values arise from the structurally impossible combination of good credit and a failed check.

6 Relation with Actual Causality

PCI establishes a conditional correspondence with actual causality: under stated assumptions, positive kernel values recover AC conditions and, with additional selection structure, identify the actual causes while avoiding exhaustive witness enumeration.

  • Computational relationship: PCI’s Monte Carlo-estimated expectation recovers actual-causality verdicts in models too large for exact enumeration under the paper’s stated conditions.Exact AC evaluation searches exponentially many suspect and witness subsets; PCI transfers the guarantee to scalable sampled estimation.
  • Formal correspondence: Under Theorem 26’s hypotheses, PCI’s positive configurations are exactly the actual causes contained in the candidate set.The result requires the stated structural constraint excluding deterministic redundancy among candidates.
  • Formal correspondence: A positive PCI kernel value corresponds to factivity and context-sensitive necessity, the two AC conditions the kernel is designed to detect.The converse theorem reads the forward proof in reverse on the necessity factor.
  • Minimality and aggregation: PCI can recover minimality when positive scores are aggregated with cardinality-decreasing candidate weights and uniform witness weights, assuming necessity dilution.A positive score for any proper subset would contradict the larger cause’s minimality; necessity dilution excludes larger nonminimal candidates from the optimizer.
  • Scope conditions: The necessity-dilution assumption holds for several conjunctive settings but fails in disjunctive overdetermination, such as factual A ∨ B = 1.In the disjunctive case, the full set can have positive necessity while proper subsets have zero necessity.
  • Scope conditions: Maximizing population-level PCI can omit an actual cause that occurs in a particular noise context, so pointwise and population-level explanations are not interchangeable.The paper’s example shows a cause actual at one noise setting but excluded from the global arg max because it contributes only there.

7 Relation with the Probability of Actual Causality

PCI and Pearl’s probability of actual causality share a probabilistic responsibility target but use different mechanisms: Pearl averages a binary actual-causation indicator, whereas PCI averages a graded necessity–sufficiency kernel over candidate causes, witnesses, alternatives, and noise. In the desert-traveller examples, both rank causes consistently, while PCI additionally captures graded responsibility, path reliability, and cross-scenario reliability.

  • Pearl’s probability of causality: Pearl’s posterior is the probability that the observed evidence corresponds to a noise state satisfying an actual-causation predicate.The predicate may use Pearl’s causal-beam definition or Halpern–Pearl’s AC1–AC3 conditions.
  • Basic comparison: PCI and Pearl agree on the cause ranking in each forensic scenario, with PCI’s J strictly larger for the cause identified by Pearl.This agreement holds at each forensic posterior.
  • Basic comparison: PCI is graded whereas Pearl is binary: Pearl’s probability saturates at 0 or 1, while PCI’s witness averaging produces intermediate magnitudes.The best witness configuration can give J = 1, but Γ-averaging dilutes the overall score.
  • Weak-poison variant: PCI assigns 5/24 ≈0.21 to a dormant cause because some witness configurations make its removal flip the outcome, information Pearl’s binary indicator discards.The non-cause suspect is marginalized rather than artificially pinned to the forensic scenario.
  • Computational distinction: PCI replaces AC’s existential search over witness and alternative pairs with a Γ ⊗ PU expectation over a graded kernel that can be estimated by sampling.AC may require checking up to 2^n witness subsets per noise state, and Pearl’s posterior inherits that enumeration.
  • Weak-poison variant: Adding stochastic fatality lowers the poisoner’s score from J = 1/3 to J = 5/24 through Γ re-weighting, not loss of sufficiency.The poisoner’s sufficiency factor remains exactly 1, while the witness pool expands from four to eight subsets.
  • Weak-poison variant: Within scenarios, PCI preserves the operative cause’s lead while distinguishing dormant responsibility: Jpoisoner = 5/24 > Jshooter = 7/48 in Scenario A, and Jshooter = 1/2 > Jpoisoner = 1/8 in Scenario B.The dormant poisoner score records marginalization over worlds where the shooter did not act either.
  • Weak-poison variant: Across scenarios, PCI ranks the reliable shooter at J = 1/2 above the unreliable poisoner at J = 5/24, although both can reach J = 1 under their best witness configuration.The difference reflects how many witness configurations attain the maximum, not different attainable ceilings.

8 Discussion and Conclusions

PCI combines witness-based causal reasoning with probabilistic sampling to produce graded, context-sensitive explanations for probabilistic causal models. The framework recovers established actual-causality results, differs structurally from SHAP-family methods, and extends to larger models while retaining important scope limitations.

  • Relationship to actual causality: The support of PCI’s causal-impact distribution recovers actual-causality verdicts under matching choices, including subset-minimal causes.Theorems 26–27 establish factivity and context-sensitive necessity, while Proposition 32 recovers AC verdicts exactly.
  • Framework: PCI integrates witness sets, alternative-value distributions, and impact functions into a probabilistic generalization of actual-causality reasoning.Witnesses preserve context sensitivity, while sampling replaces exhaustive enumeration of suspect sets and counterfactual alternatives.
  • Comparison with attribution methods: PCI uses intervention with witnesses rather than observational marginalization, avoiding characteristic SHAP failures on overdetermination and continuous mediation.The same witness mechanism also supports the upstream-versus-downstream attribution split in the production case study.
  • XAI desiderata: PCI satisfies sparsity, multiple-counterfactual, and causal-constraint desiderata, but only partially satisfies model-agnosticism because black-box wrapping sacrifices causal sensitivity.Small suspect sets bias explanations toward sparse causes, and causal-model interventions propagate changes to descendants by construction.
  • Limitations: PCI requires a trained probabilistic causal model and inherits systematic attribution errors from misspecified structural equations or noise distributions.The method does not itself flag model misspecification from attribution scores alone.
  • Limitations: Fixed-variance Monte Carlo estimation still scales with suspect and witness-set cardinalities, and the paper lacks a general rule for knowing when fewer samples suffice.The production case is presented as a feasibility witness rather than quantitative evidence for an apples-to-apples SHAP comparison.

A Computations for the OBCB Running Example

The OBCB computation traces Alice’s counterfactual loan outcome after intervening on gender while holding her bad credit fixed. The calculation separates checked and unchecked branches through the model’s stochastic mechanisms.

  • Counterfactual setup: Alice’s factual configuration is female, bad credit, and loan denial; the counterfactual sets gender to male while retaining bad credit.The intervention is evaluated by tracing the model’s downstream check and loan mechanisms.
  • Branch calculations: Under the intervention, men are checked with probability 0.9, and a checked applicant with bad credit receives a loan with probability 0.05.If the applicant is not checked, the model deterministically denies the loan.
  • Branch calculations: The computation combines the checked and unchecked branches to obtain the counterfactual outcome probability.The appendix presents the branch traces underlying Alice’s individual probability calculations.

A.2 Alice’s probability of sufficiency

Alice’s probability-of-sufficiency calculation conditions on her bad credit and on the counterfactual state of male gender with loan approval. Restoring female gender then yields loan denial in both possible checking branches.

  • Conditional setup: The individual probability-of-sufficiency calculation fixes Alice’s bad credit while flipping gender to male and the outcome to loan approval.The intervention restores female gender and asks whether the loan is denied again.
  • Noise coupling: Conditioning on male gender and loan approval restricts the shared noise to cases where the male applicant was checked, giving female checking probability 2/9.This differs from the unconditional female checking probability of 0.2 because the calculation uses a twin-network coupling.
  • Result: With female gender and bad credit, both the checked and unchecked branches deny the loan, so the conditional probability of denial is 1.The resulting expression is P(loan_gender=female = F | gender = male, loan = T, credit = bad) = 1.

A.3 SHAP value function for Option A

This appendix defines the SHAP coalition value function and applies it to the OBCB and continuous mediation examples. It contrasts observational and causal conditioning choices while retaining the standard Shapley axioms and deriving model-specific coalition quantities.

  • OBCB value function: The OBCB SHAP value function assigns coalition values by averaging model outputs over dropped features under the stated feature-dropping convention.For the two-feature game, 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.
  • OBCB value function: SHAP values are obtained by applying the closed-form two-player Shapley formula to coalition values and flipping signs when attributing rejection.The resulting magnitudes are reported in the appendix’s table.
  • Continuous mediation example: For the Gaussian chain X → M → Y, the appendix derives variances, covariances, conditional expectations, and coalition values for targets Y, M, and X.The model uses X∼N(0.5,0.25), additive Gaussian noise with variance 0.1, and factual values X*=M*=Y*=1.
  • SHAP definition: The appendix distinguishes SHAP coalitions from PCI suspect sets and defines the characteristic function for feature subsets held at factual values.The standard Shapley axioms include symmetry, dummy, and linearity, alongside the usual formula.
  • Observational and causal SHAP: The original SHAP formulation conditions dropped features on retained factual features, while an alternative averages them independently under marginal distributions.The appendix also gives a causal variant that uses interventional distributions for dropped features affected by retained features.
  • Continuous mediation example: The continuous example uses conditional expectations to evaluate coalitions, including v({Y})=0.778 for target M and v({M,Y})=0.857.These quantities exploit the conditional independence X ⟂ Y | M in the mediation chain.

B.5 Causal SHAP

The Causal SHAP examples show that causal interventions can correct some feature-attribution errors, but still reproduce plain SHAP when causal and correlational valuations coincide.

  • The comparison therefore illustrates that causal structure changes attribution for some targets but not when intervention values coincide with plain SHAP.
  • For target Y, Causal SHAP matches plain SHAP, assigning ϕX = ϕM = 0.25.All coalition values are identical under both methods.
  • For target M, Causal SHAP assigns vcausal({Y}) = 0.75, below plain SHAP’s 0.889 because Y does not cause X.
  • For target X, Causal SHAP assigns vcausal({Y}) = 0.5, below plain SHAP’s 0.778 because Y is not a cause of M.

B.6 PCI

PCI evaluates causal responsibility by sampling counterfactual necessity and sufficiency worlds across candidate causes and witness sets. Its examples use explicit interventions, alternative-value distributions, and Monte Carlo or matching enumeration.

  • PCI scores a suspect with ci(ys, yn, y⋆) = |yn − y⋆| − |ys − y⋆| using fresh exogenous noise in necessity and sufficiency worlds.
  • For R(X ⇝Y), PCI samples X′ ∼N(0.5, 0.25) and averages accepted candidate-cause and witness pairs, including a witness that pins M at its factual value.
  • The closed-form computation gives R(X ⇝Y) ≈0.186, while R(Y ⇝X | W={M}) = 0 because X is exogenous.
  • The desert-traveller comparison runs PCI’s Monte Carlo search on scenario-specific structural models and checks its results against hand-rolled enumeration within Monte Carlo noise.
  • In that comparison, suspects are {X, P}, witnesses are mediators {c, d}, and the non-cause suspect follows the structural model rather than being hard-coded.
  • The weak-poison variant expands the witness pool to {c, d, vC} and changes the forensic posterior pins for the scenario variables.

D Synthetic Evaluation Across Linear, Overdetermined, and Undercutting Archetypes

The synthetic evaluation tests PCI on linear, overdetermined, preempted, and irrelevant-variable archetypes with analytic expectations and bootstrap-derived thresholds. PCI confirms the qualitative desiderata, while two quantitative preemption diagnostics remain unresolved or mismatched.

  • Model: The synthetic model combines linear, overdetermined, preempted, and irrelevant-variable branches feeding one outcome through deterministic mediators.
  • Model: D is sampled but has no path into E, so its non-zero PCI score estimates the procedure’s noise floor.
  • Evaluation design: The two factual cases vary the gate and the separation between overdetermined contributors to test whether PCI tracks structural role rather than variable identity.
  • Results: At N = 20 000 samples, the bootstrap-derived threshold is ϵ ≈0.151, and 10 of 12 desiderata survive it.
  • Results: All ten forest-plot claims fall inside their pass bands, with the tightest claim showing ∆N(P) = +0.250 against ϵ ≈0.151.
  • Diagnostics: The cross-regime P flip is statistically unresolved, while P fails to quantitatively match L1 in Case 2; the supported conclusion is only that both clear the noise floor.
  • Transition: The evaluation complements the archetype ground-truth tests with a continuous-input comparison against gradient-based attribution.

E Comparison to Differential Causal Effect

PCI is compared with Differential Causal Effect, which measures local response gradients rather than individual-level responsibility. In a credit-limit model, PCI consistently ranks age above application time and remains stable across time-unit reparameterization, unlike DCE.

  • Methods: DCE is a gradient-based rate-of-change measure, whereas PCI estimates individual-level responsibility from necessity- and sufficiency-world densities.
  • Methods: PCI requires only a sampleable causal model, while DCE requires a differentiable function and produces unit-sensitive cross-feature comparisons.
  • Credit-limit example: In the credit-limit example, PCI remains informative in flat response regions because it depends on counterfactual densities rather than only local gradients.
  • Caveat: The unit-invariance result depends on specifying the alternative-value distribution in standardized or probability-mass terms rather than with a fixed raw-unit distance.
  • Results: PCI ranks age above time of application in every setting, with age scores roughly 3/2 as large and hours-versus-minutes scores essentially unchanged.
  • Results: Across the factual grid, PCI favors age everywhere, while DCE favors application time over most of the hour-scale grid and changes sharply when time is measured in minutes.

F Scaling Actual-Cause Computation: An Empirical Comparison

The benchmark compares exact actual-cause enumeration with a Monte Carlo PCI estimator on a structured, scalable throwing problem. PCI evaluates far fewer configurations while retaining high attribution accuracy across substantially larger problem sizes, although the benchmark’s structure limits the conclusion.

  • Benchmark setup: The benchmark replicates the canonical throwing problem across n sites, preserving overdetermination, undercutting, witness-mediated symmetry-breaking, and a closed-form ground truth.The model contains 4n + 1 variables, with roots as suspects and deterministic mediators as witnesses.
  • Exact method: The exact procedure enumerates non-empty cause-witness pairs and reaches 9, 225, 3969, and 65025 pairs at n = 1, 2, 3, and 4.Each candidate cause and witness subset is intervened on, and minimality is checked against proper cause subsets.
  • Approximate method: PCI replaces exhaustive enumeration with Monte Carlo draws that sample suspect and witness sets, then compare necessity and sufficiency worlds under shared noise.The estimator records counterfactual outcome changes and aggregates per-suspect scores from sampled configurations.
  • Results: The exact method times out at 4n + 1 = 17 variables, whereas approximate methods reach roughly 73–145 variables depending on sample budget.The frugal α=0.5 run reaches the largest tested sizes.
  • Results: Approximate methods visit at most a few hundred configurations per problem size, creating a several-orders-of-magnitude gap that widens with n.The exact method visits the full powerset.
  • Accuracy and scope: Witness-using runs remain above 0.85 correct-attribution rate through 73–89 variables, while matched-budget witness runs average 0.941 versus 0.861 without witnesses.Witness runs lead at 22 of the 26 shared problem sizes, but the benchmark uses structured responsibility and witness conditions.

G Dynamical SIR Benchmark: PCI on Continuous Outcomes

The SIR benchmark tests PCI on a Bayesian dynamical model with a continuous overshoot outcome and interacting policies. PCI identifies lockdown as the dominant and robust cause, while masking is context-sensitive despite similar but-for probabilities.

  • Model and query: The benchmark uses a Bayesian SIR model with continuous outcomes, stochastic transmission and recovery rates, and interacting lockdown and masking policies.Lockdown efficiency is 0.6, while masking efficiency depends on whether lockdown is present.
  • Model and query: But-for probabilities are similar for lockdown-only and mask-only regimes: 0.84 versus 0.81 for Pr(overshoot > 24), so but-for analysis cannot distinguish them at this sample budget.The comparison uses 100 predictive draws per policy regime.
  • PCI procedure: PCI samples 2,500 intervention regimes per suspect and scores necessity and sufficiency through |y⋆−ynec| − |y⋆−ysuff|.The suspect set contains lockdown and mask; policy efficiencies serve as witnesses, excluding the focal suspect from the witness set.
  • Single-world results: Lockdown ranks above mask because removing it moves overshoot farther from y⋆: mean absolute necessity distances are 8.276 versus 7.536, accounting for 0.740 of the 0.835 total gap.The sufficiency terms differ by only 0.095, and the reported score uses q = 0.754 with q^2 scaling.
  • Robustness: Across 20 factual worlds, lockdown averages +0.819 versus mask’s +0.118, leads in 18 worlds, and has a mean gap of +0.701 with standard error 0.111.The worlds span factual overshoots from 9.86 to 33.80 people.
  • Context sensitivity: Lockdown remains positive across context regimes, whereas mask changes sign, showing that PCI distinguishes robust causes from context-sensitive ones.The different-contexts experiment fixes or frees the partner-policy efficiency variable as a witness.

H Scaling PCI to a Real-World Automated Valuation Model

The paper applies PCI to a deployed residential valuation model using a disclosed New York prototype and a larger production system. PCI is computationally feasible at production scale and yields attribution patterns that differ qualitatively from SHAP, but quantitative accuracy cannot be assessed there.

  • Scope and limitation: The real-world study provides no quantitative accuracy comparison because disclosure constraints prevent access to the trained model and training data.Its role is limited to production-scale feasibility and qualitative SHAP-versus-PCI divergence; controlled synthetic benchmarks provide the quantitative validation.
  • Setting: The deployed study applies PCI to a residential property valuation model, using a New York prototype that shares the full system’s architecture and causal-kernel construction.The prototype is smaller than the undisclosed national model.
  • Causal model: The prototype DAG has 25 nodes and 99 edges, while the full national DAG is larger and exact actual-causality enumeration is computationally infeasible.The graph was manually constructed with domain experts and refined using data-supported edges.
  • Causal model: The causal model represents variables with normal or categorical causal kernels parameterized by neural networks, with the outcome modeled by a sparse variational Gaussian process.Training uses cross-entropy for categorical distributions and average negative log likelihood for continuous distributions.
  • Feasibility: PCI estimates feature impact by Monte Carlo sampling from the expanded model, with estimation converging at around 25,000 samples.Scores are conditioned on each suspect being active and averaged to estimate its expected impact.
  • Comparison to SHAP: On the trained model, SHAP concentrates attribution on downstream, high-in-degree variables, whereas PCI distributes attribution more evenly across the upstream causal structure.The comparison concerns attribution-shape divergence rather than the substantive content of the assigned features.
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