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Cyclic Counterfactuals under Shift–Scale Interventions
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Theoretical framework for counterfactual inference in cyclic SCMs under shift–scale interventions, showing global contraction implies simplicity, twin-SCM well-posedness, closure under composition, and sub-Gaussian counterfactual tails.
Problem
Counterfactual inference in cyclic structural causal models is ill-posed in general because cyclic structural equations may lack unique solutions; soft shift–scale interventions need well-posedness guarantees.
Approach
Uses a global l^p contraction condition on the SCM mechanism to guarantee unique solvability (simplicity) for all subsets, shows the twin SCM stays contractive under shift–scale interventions with |a_j|≤1, and derives Gaussian-noise sub-Gaussian concentration for counterfactual functionals.
Key results
- Proves global l^p contraction (constant kappa<1) implies unique solvability w.r.t. every subset, hence simple SCM, even with cycles.
- Shows the shift–scale intervened twin SCM remains uniquely solvable when all scale coefficients satisfy |a_j|≤1, and that composed shift–scale interventions stay kappa-contractive.
- Derives sub-Gaussian tail bound with proxy 2(1-kappa)^{-2}sigma^2 for 1-Lipschitz functionals of counterfactuals under Gaussian noise; a linear 2-variable consumption–income example is worked out in closed form.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply a shift–scale intervention to income with α=0.8 and β=1.0 | Linear cyclic SCM example · Linear cyclic SCM example | mean consumption: 1.5625 -> 2.024 | Helped |
| Apply a shift–scale intervention to income with α=0.8 and β=1.0 | Linear cyclic SCM example · Linear cyclic SCM example | mean income: 1.125 -> 2.048 | Helped |
| Apply a shift–scale intervention to income with α=0.8 and β=1.0 | Linear cyclic SCM example · Linear cyclic SCM example | C–I correlation: 0.75 to 0.69 | Mixed |
| Impose a global ℓp-contraction condition | Cyclic SCMs · Cyclic SCMs with closed coordinate domains | unique solvability | Helped |
| Apply shift–scale interventions with |a_j|≤1 | Twin SCM of a globally contractive SCM · Intervened twin SCM | unique solvability | Helped |
| Compose finitely many shift–scale interventions, each with |a_j|≤1 | Globally contractive SCM · Composed shift–scale interventions | contraction constant: κ | No clear effect |
| Use scale factors with |a_j|>1 without verifying κ_max<1 | Shift–scale intervention on a globally contractive SCM · Scale factors exceeding one | unique solvability: κ_max < 1; κ_max≥1 | Mixed |
Limitations
- Requires global contraction condition uniformly over the state space, which may fail in some feedback systems.
- Concentration result relies on Gaussian exogenous noise; heavy-tailed noise gives only polynomial concentration.
- Closure results cover only shift–scale maps with |a_j|≤1; larger scale factors need extra conditions.
Experimental setup
Builds on: Pearl's abduction-action-prediction twin network counterfactuals simple SCMs (Bongers et al.) Banach fixed-point theorem
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