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This paper introduces two families of cost-aware stochastic interventions for discrete treatments, connecting causal modeling with cost-sensitive decision making. The core idea is to define these interventions as marginals of a bivariate distribution that solves a cost-penalized information projection (CPIP) of the independent product of organic propensity scores and a reference policy. This yields closed-form Boltzmann-Gibbs couplings, whose induced marginals smoothly interpolate, via a tilt parameter δ, from the organic law or a reference law towards a product-of-experts limit. The first family generalizes incremental propensity score interventions (IPIs) by incorporating explicit cost structures and target policies, while retaining identification without global positivity. For inference, the paper derives efficient influence functions under a nonparametric model and constructs one-step estimators. Empirically, simulations demonstrate that the proposed estimators improve stability and robustness to nuisance misspecification compared to plug-in baselines.

: Classical causal inference often uses hard interventions, which deterministically set exposure to a fixed value. This approach is overly rigid for many real-world domains, such as healthcare and economic policy, where information may be incomplete, treatment allocation is resource-constrained, or inference is unstable due to zero probabilities for certain treatments in subpopulations.

Introduction of two families of stochastic interventions with discrete treatments that connect causal modeling to cost-sensitive decision making.
Formulation of these interventions as marginals of a bivariate distribution solving a cost-penalized information projection (CPIP) of the independent product of organic propensity scores and a reference policy.
Derivation of closed-form Boltzmann-Gibbs couplings, whose induced marginals define modified stochastic policies that interpolate smoothly, via a tilt parameter δ, from the organic law or a reference law toward a product-of-experts limit when all destination costs are strictly positive.
Generalization and extension of incremental propensity score interventions (IPIs) to incorporate explicit cost structures and target policies, while retaining identification without global positivity.
Derivation of efficient influence functions (EIFs) under a nonparametric model for inference on expected outcomes after these policies.
Development of one-step estimators that improve stability and robustness to nuisance misspecification relative to plug-in baselines in simulations.
Construction of asymptotically valid confidence intervals and uniform confidence bands over a specified segment of the interpolation range.

The method introduces two families of stochastic interventions derived from a Cost-Penalized I-Projection (CPIP). CPIP Objective: Given two input probability measures π (source) and ν (target) over an action set A, a summable/integrable cost function c: A² → [0, ∞), and a penalization parameter δ ≥ 0, the CPIP of the independent product π ⊗ ν is defined as the joint distribution γ ∈ M(A²) that solves: inf DKL(γ || π ⊗ ν) + δ Eγ {c(A', A")} where DKL is the Kullback-Leibler divergence. Key Equations (Boltzmann-Gibbs Kernel): This problem admits a closed-form solution, with its unique minimizer given by the Boltzmann-Gibbs kernel: γδ(a', a") ∝ π(a') ν(a") e^(-δc(a',a")) Tilted Marginal Distributions: The two families of stochastic interventions are defined as the marginal distributions of this CPIP solution. For a categorical point-exposure variable A, a target distribution ν(·|w), and propensity scores π(·|w) given covariates W=w, with cost c(a', a") of reallocating a unit from a' to a", the tilted source marginal πδ(a|w) and tilted target marginal νδ(a|w) are: πδ(a|w) := (π(a|w) Σa''∈A ν(a''|w) e^(-δc(a,a'')) ) / (Σa',a''∈A π(a'|w) ν(a''|w) e^(-δc(a',a'')) ) νδ(a|w) := (ν(a|w) Σa'∈A π(a'|w) e^(-δc(a',a)) ) / (Σa',a''∈A π(a'|w) ν(a''|w) e^(-δc(a',a'')) ) These marginals interpolate smoothly via the tilt parameter δ. When δ=0, they recover the input distributions. As δ → ∞, they approach a product-of-experts (PoE) limit if all destination costs are strictly positive. Estimation and Inference: For estimating mean outcomes after these interpolated policies, the paper derives efficient influence functions (EIFs) in a nonparametric model. One-step estimators are constructed by substituting nuisance estimates (π and Q) into the EIFs. Inference uses sample splitting and cross-fitting to relax technical conditions for weak convergence, estimating nuisance functions on one subset and evaluating the influence function on a disjoint subset. Uniform confidence bands are constructed using a multiplier bootstrap.

Intervention Definition: Two families of stochastic interventions are introduced, indexed by a single tilt parameter δ.
Cost Function: c: A² → [0, ∞) is a summable/integrable cost function on pairs of actions (A', A").
Penalization Parameter: δ ≥ 0.
Key Equations (CPIP):
- Objective: inf DKL(γ || π ⊗ ν) + δ Eγ {c(A', A")}
- Boltzmann-Gibbs Kernel: γδ(a', a") ∝ π(a') ν(a") e^(-δc(a',a"))
Key Equations (Tilted Marginals):
- πδ(a|w) = (π(a|w) Σa''∈A ν(a''|w) e^(-δc(a,a'')) ) / (Σa',a''∈A π(a'|w) ν(a''|w) e^(-δc(a',a'')) )
- νδ(a|w) = (ν(a|w) Σa'∈A π(a'|w) e^(-δc(a',a)) ) / (Σa',a''∈A π(a'|w) ν(a''|w) e^(-δc(a',a'')) )
- For Remark 2 (treatment-specific costs):
  - ξδ(a) := ν(a) (1 – e^(-δc(a)))
  - ζδ := Σa'∈A ν(a') e^(-δc(a'))
  - πδ(a|w) := (ζδ + ξδ(a)) π(a|w) / (Σa'∈A (ζδ + ξδ(a')) π(a'|w))
  - νδ(a|w) := (ν(a) – ξδ(a)(1 – π(a|w))) / (Σa'∈A (ζδ + ξδ(a')) π(a'|w))
Identification: Expected outcomes µδ^S and µδ^T are identified from observational data under conditional ignorability/backdoor admissibility of W (or a smaller set Z ⊆ W).
- µδ^S = Σa∈A Ew {πδ(a | W)Q(Z,a)}
- µδ^T = Σa∈A Ew {νδ(a | W)Q(Z,a)}
Estimation: Plug-in estimators for µδ^S and µδ^T are obtained by estimating π and Q, applying transformations (7) and (8), and averaging over i.i.d. samples.
Efficient Influence Functions (EIFs): Uncentered EIFs for Sδ[P] and Tδ[P] are derived for a nonparametric model.
- Dδ^S(Oi) = Dδ^S,1(Oi) + Dδ^S,2(Oi)
- Dδ^S,1(Oi) = (Y_i - Q(Z_i, A_i)) / (π(A_i | W_i)) * πδ(A_i | W_i)
- Dδ^S,2(Oi) = Σa∈A πδ(a | W_i) Q(Z_i, a)
- Dδ^T(Oi) = Dδ^T,1(Oi) + Dδ^T,2(Oi) + Dδ^T,3(Oi)
- Dδ^T,1(Oi) = (Y_i - Q(Z_i, A_i)) / (π(A_i | W_i)) * νδ(A_i | W_i)
- Dδ^T,2(Oi) = (2 - νδ(A_i | W_i) / π(A_i | W_i)) * Σa∈A νδ(a | W_i) Q(Z_i, a)
- Dδ^T,3(Oi) = (νδ(A_i | W_i) / π(A_i | W_i)) * ρδ(A_i) Q(Z_i, A_i) - Σa∈A (π(a | W_i) / π(A_i | W_i)) * ρδ(a) Q(Z_i, a)
  - where ρδ(a) := ξδ(a) / (ζδ + ξδ(a))
Optimization: One-step estimators are Newton-Raphson corrections.
Inference: Sample splitting and cross-fitting are used. Nuisance functions (π and Q) are fitted on training folds, and EIFs are evaluated on held-out folds.
Confidence Bands: Uniform confidence bands are constructed using a multiplier bootstrap with B i.i.d. multipliers x_i^(b) ~ N(0, 1).

Datasets:
- Simulations: Finite synthetic data adapted from Kang and Schafer (2007) and Kennedy (2018), with a three-leveled categorical exposure and increased noise.
- Application Case: Linked data from national registries for 8,609 Norwegian children born between 2000 and 2007, diagnosed with ADHD, focusing on numeracy test scores in grade 8th.
Metrics:
- Integrated bias (iBias)
- Integrated root mean squared error (iRMSE)
Baselines: Plug-in estimators.
Numeric Results (Simulations):
- Results are averaged over 200 Monte Carlo repetitions with n = 1000.
- One-step estimators vs. Plug-in: Across all three setups (different cost functions c and target distributions ν), one-step estimators consistently outperform plug-in estimators.
- Correct Specification: Under correct specification, one-step estimators achieve essentially zero bias (e.g., 0.02 for πδ and 0.57 for νδ in Setup 1) and lower or comparable RMSE (e.g., 2.20 for πδ and 8.44 for νδ in Setup 1) compared to plug-in (e.g., 0.40 iBias for πδ, 3.03 iBias for νδ).
- Outcome Model Misspecification (Q): One-step estimators retain low bias and RMSE (e.g., 0.03 iBias for πδ, 0.35 iBias for νδ in Setup 1) while plug-in estimators incur substantial deterioration (e.g., 1.99 iBias for πδ, 15.22 iBias for νδ in Setup 1).
- Propensity Score Misspecification (π): One-step estimators still deliver lower bias and RMSE (e.g., 0.39 iBias for πδ, 3.98 iBias for νδ in Setup 1) than plug-in (e.g., 14.00 iBias for πδ, 7.92 iBias for νδ in Setup 1).
- Sensitivity: µδ^T is more sensitive to misspecification than µδ^S, but the one-step correction attenuates this sensitivity.
Numeric Results (Application Case):
- Estimated expected outcomes increase gradually with δ, indicating a small benefit of less than one test score point.
- Across most of the range of δ ∈ [−1, 2], the target-tilted policy νδ achieves consistently higher expected outcomes than the source-tilted policy πδ.
- Target distributions placing greater emphasis on low-dose treatment are associated with better academic achievement outcomes.
Runtime/Training Cost: Not stated in the paper.

Negative Tilt Parameter (δ < 0): For δ < 0, the joint distribution γ in (4) is not the optimizer of the CPIP program. While the associated Boltzmann-Gibbs kernel is summable for finite action sets and bounded costs, delivering smooth parametric extensions, for continuous spaces with potentially unbounded costs, additional integrability or tail conditions are required for δ < 0 to deliver well-posed stochastic policies.
Tilted Target Marginal Degeneracy (Remark 1): In the specific setting where the target marginal ν is a degenerate distribution that always assigns treatment (e.g., ν(a|w) = I(a = 1)) and the cost is Hamming cost (c(a', a") = I(a' ≠ a")), the tilted target marginal νδ(a|w) coincides with the input target distribution and does not yield a differentiated stochastic policy for δ > 0.
Pushforward Distribution vs. CPIP Marginals: The CPIP criterion optimizes over joint laws without imposing or penalizing any marginal constraint, so the tilted marginal νδ(w) is simply the second marginal of γ and does not represent the distribution obtained by pushing π(·|w) through γ. A systematic comparison between CPIP marginals, marginal-penalized optimal transport variants, and pushforward distributions is left for future work.
Global Positivity for Tilted Target Policy: For the expected outcome after the tilted target policy µδ^T to be well defined and estimable, a global positivity condition is required: sup_a∈A (ν(a | W) / π(a|W)) < ∞, Pw-almost surely. This is not required for µδ^S.

Method evidence: "We introduce two families of stochastic interventions with discrete treatments that connect causal modeling to cost-sensitive decision making. The interventions arise from a cost-penalized information projection of the independent product of the organic propensity scores and a reference policy, yielding closed-form Boltzmann-Gibbs couplings." (section: Abstract)
Result evidence: "Across all three setups, the proposed one-step estimators consistently outperform the corresponding plug-in estimators. Under correct specification, the one-step estimators attain essentially zero bias and achieve lower or comparable RMSE, with the gains most pronounced after the policy πδ." (section: Results)
Limitation evidence: "For δ < 0, the joint distribution γ in (4) is not the optimizer of the CPIP program; however, when the action set A is finite and costs are bounded, the associated Boltzmann-Gibbs kernel is summable, so γ remains normalizable and produces smooth and closed-form parametric extensions for πδ and νδ. In this extrapolative regime, the induced tilted marginals shift mass toward higher-cost and repulsive pairings and are pushed away from the independence law π ⊗ ν, favoring more adversarial couplings. For continuous spaces with potentially unbounded costs, additional integrability or tail conditions are required for δ < 0 to deliver well posed stochastic policies." (section: Tilted Marginal Distributions)

Interpolated Stochastic Interventions Based on Propensity Scores, Target Policies and Treatment-Specific Costs

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