Technical Summary: DeepSWIP – Quotient-WMC Counterfactuals for Neural Probabilistic Logic Programs

1. Problem Statement

Neurosymbolic systems like DeepProbLog effectively combine neural perception with probabilistic logic to handle perceptual uncertainty. However, standard inference in these systems is associational (computing $P(Y|X=x)$) rather than causal. They cannot natively answer counterfactual queries (e.g., "What would have happened if $X$ were $x$, given evidence $E=e$?") which require a causal semantics for interventions ($do(X=x)$).

Existing approaches to counterfactuals in Probabilistic Logic Programming (PLP) often rely on Twin Networks, which duplicate the program structure into factual and counterfactual copies while sharing exogenous choices. While semantically convenient, this duplication is computationally expensive for relational programs and awkward for neural predicates. Specifically, it is unclear whether neural predicates (e.g., convolutional networks) should be evaluated twice (once per world) or shared as exogenous uncertainty. Furthermore, duplication obscures the single-world semantics of interventions, creating unnecessary independence assumptions.

2. Methodology: DeepSWIP

The paper introduces DeepSWIP (Deep Single World Intervention Programs), a framework that extends the SWIP methodology (Habib et al., 2026) to DeepProbLog. The core idea is to separate neural perception from symbolic counterfactual inference by treating neural predicates as fixed probabilistic choices before applying causal transformations.

2.1 Neural Materialization

For a fixed set of neural parameters $\eta$ and input contexts $c$, a neural predicate $nn(m, x, y)$ is evaluated once. Its output distribution over symbolic values is "materialized" into ordinary probabilistic facts (annotated disjunctions) within the logic program.

• Transformation: The neural network is frozen, and its output probabilities $p_{k,j}$ replace the neural predicate with categorical choices: $p_{k,1} :: a_k(y_{k,1}); \dots; p_{k,r_k} :: a_k(y_{k,r_k})$.
• Result: The DeepProbLog program is converted into an ordinary ProbLog program ($P_{mat}$) where all uncertainty is represented by independent probabilistic facts.

2.2 Single-World Intervention Surgery

Once materialized, the program undergoes SWIP transformation (Single World Intervention Program) to handle the intervention $do(X=x)$:

1. Deletion: All rules defining the intervened atom $X$ are removed.
2. Redirection: Downstream rules that previously depended on $X$ are rewritten to depend on a fixed atom $X_{fix}(x)$.
3. Assertion: A deterministic fact $X_{fix}(x)$ is added.
   This process creates a single transformed program $\Delta_x$ that represents the counterfactual world without duplicating the structure.

2.3 Quotient-WMC Semantics

Counterfactual probabilities are computed using Weighted Model Counting (WMC) over the single transformed program. The counterfactual conditional is defined as a quotient of two WMC polynomials:
$$Q(p) = P(Y_x = y \mid E = e) = \frac{\Phi^x_{Y,E}(p)}{\Phi^x_E(p)}$$
Where:

• $\Phi^x_{Y,E}(p) = \text{WMC}(\Delta_x \land Y=y \land E=e; p)$
• $\Phi^x_E(p) = \text{WMC}(\Delta_x \land E=e; p)$
• $p$ represents the materialized neural probabilities.

This algebraic representation unifies the inference task, allowing for exact computation relative to the learned materialized Functional Causal Model (FCM).

3. Key Contributions

The paper makes four primary contributions:

1. Neural Materialization: A method to convert fixed-context DeepProbLog programs into ordinary ProbLog programs, enabling standard causal program transformations.
2. Correctness Theorem: A proof that DeepSWIP is exact relative to the learned materialized FCM, provided the program has unique supported models.
3. Quotient-WMC Analysis: A theoretical analysis revealing that:
   • Intervention Cleaning: Neural probabilities associated with intervened mechanisms have zero derivative in the counterfactual query (they are surgically removed).
   • Calibration Sensitivity: The error in counterfactuals depends on the calibration of neural probabilities, not just classification accuracy.
   • Rare-Evidence Instability: The sensitivity of the counterfactual estimate grows inversely with the probability of the evidence ($1/P(E=e)$), leading to potential instability with rare evidence.
4. Empirical Validation: Experiments on MPI3D and SUMO traffic simulations demonstrating the framework's correctness, speed, and statistical properties.

4. Experimental Results

4.1 MPI3D Visual Counterfactuals

• Setup: A visual reasoning task involving neural predicates for shape, size, and color, combined with symbolic rules for physical stability.
• Correctness: DeepSWIP agreed with a DeepTwin construction on 100% of 12,000 queries, validating the transformation.
• Efficiency: DeepSWIP achieved a 2.14× speedup (3.57 ms/query vs. 7.65 ms/query) by avoiding the endogenous duplication required by Twin Networks.
• Error Propagation: Root Mean Square Error (RMSE) was zero for queries determined solely by the intervention (demonstrating intervention cleaning) and small for queries depending on non-intervened predicates.
• Pruning: The SWIP transformation effectively pruned irrelevant clauses (e.g., reducing statement counts from 18 to 5 for specific queries).

4.2 SUMO HOV Traffic Experiments

• Setup: A traffic simulation testing calibration sensitivity and Double Machine Learning (DML) correction.
• Findings:
  • As sensor noise increased, neural calibration degraded, leading to significant bias in naive plug-in counterfactual estimates.
  • AIPW/DML: An Augmented Inverse Propensity Weighted (AIPW) estimator, using a known randomized policy, successfully removed most first-order bias for population mean and Average Treatment Effect (ATE) estimands, even with degraded neural calibration.
  • This highlights that while symbolic inference is exact relative to the materialized probabilities, statistical estimation of those probabilities remains a source of error.

5. Significance and Claims

The paper positions DeepSWIP not as a universal causal discovery method, but as a precise counterfactual inference layer for neurosymbolic programs with explicit causal structures and learned probabilistic mechanisms.

• Theoretical Insight: The quotient-WMC view provides a unified explanation for why intervention cleaning works, why calibration is more critical than accuracy for counterfactuals, and why rare evidence causes instability.
• Practical Impact: By reducing counterfactual inference to a single-world WMC problem, DeepSWIP offers a computationally efficient alternative to Twin Networks, particularly for programs with neural predicates.
• Limitations: The authors explicitly note that the exactness of DeepSWIP is relative to the learned materialized FCM. If neural predicates are miscalibrated, the symbolic counterfactual will be "exactly wrong" relative to the true data-generating process. The framework separates exact symbolic inference from the statistical error of estimating neural probabilities.

The work concludes that DeepSWIP successfully bridges the gap between neural perception and single-world causal reasoning, offering a robust foundation for counterfactual analysis in neurosymbolic AI.