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Interpolated stochastic interventions based on propensity scores, target policies and treatment-specific costs

Johan de Aguas

TL;DR

This work develops cost-aware stochastic interventions for discrete treatments by formulating a cost-penalized information projection that yields Boltzmann-Gibbs couplings and tilted marginals. The two delta-indexed families interpolate between input policies and a product-of-experts limit under positive costs, controlled by a single tilt parameter. Efficient semiparametric estimators based on influence functions and one-step corrections are derived, with cross-fitting and uniform confidence bands for inference. The framework supports graded hypothesis testing and policy design under budgets, enabling prospective policy prototyping from observational data prior to experiments.

Abstract

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. The induced marginals define modified stochastic policies that interpolate smoothly, via a tilt parameter, from the organic law or from the reference law toward a product-of-experts limit when all destination costs are strictly positive. The first family recovers and extends incremental propensity score interventions, retaining identification without global positivity. For inference on the expected outcomes after these policies, we derive the efficient influence functions under a nonparametric model and construct one-step estimators. In simulations, the proposed estimators improve stability and robustness to nuisance misspecification relative to plug-in baselines. The framework can operationalize graded scientific hypotheses under realistic constraints. Because inputs are modular, analysts can sweep feasible policy spaces, prototype candidates, and align interventions with budgets and logistics before committing experimental resources.

Interpolated stochastic interventions based on propensity scores, target policies and treatment-specific costs

TL;DR

This work develops cost-aware stochastic interventions for discrete treatments by formulating a cost-penalized information projection that yields Boltzmann-Gibbs couplings and tilted marginals. The two delta-indexed families interpolate between input policies and a product-of-experts limit under positive costs, controlled by a single tilt parameter. Efficient semiparametric estimators based on influence functions and one-step corrections are derived, with cross-fitting and uniform confidence bands for inference. The framework supports graded hypothesis testing and policy design under budgets, enabling prospective policy prototyping from observational data prior to experiments.

Abstract

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. The induced marginals define modified stochastic policies that interpolate smoothly, via a tilt parameter, from the organic law or from the reference law toward a product-of-experts limit when all destination costs are strictly positive. The first family recovers and extends incremental propensity score interventions, retaining identification without global positivity. For inference on the expected outcomes after these policies, we derive the efficient influence functions under a nonparametric model and construct one-step estimators. In simulations, the proposed estimators improve stability and robustness to nuisance misspecification relative to plug-in baselines. The framework can operationalize graded scientific hypotheses under realistic constraints. Because inputs are modular, analysts can sweep feasible policy spaces, prototype candidates, and align interventions with budgets and logistics before committing experimental resources.

Paper Structure

This paper contains 21 sections, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and Contribution
  3. Preliminaries
  4. Hard Interventions
  5. Incremental Propensity Score Interventions (IPI)
  6. Cost-Penalized I-Projection
  7. Tilted Marginal Distributions
  8. Pushforward Distribution
  9. Identification
  10. Semiparametric Efficient Estimation
  11. Inference
  12. Simulations
  13. Application Case
  14. Conclusions
  15. Acknowledgments
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Tilted source distributions $\pi^\star_\delta$ in panels (a) and (b), and tilted target distributions $\nu^\star_\delta$ in panels (c) and (d), for a binary exposure. Curves show pointwise transformation of the propensity score $\pi(1\,|\, w)$ for $\delta\in\{1.0,2.5\}$. Target configuration is encoded by dot pattern: $\nu=(0,1)$ is non-dotted and $\nu=(0.7,0.3)$ is dotted. Cost structure is encoded by line style: solid for $c=(1,1)$ and dashed for $c=(0,2)$. Vector components are ordered as $(A=0, A=1)$.
  • Figure 2: Estimated expected outcome after stochastic interventions $\pi_\delta^\star$ and $\nu_\delta^\star$ in the application case with 3-levels exposure: (1) no treatment, (2) low dose, and (3) high dose treatment. Broken lines represent uniform confidence bands.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • proof
  • proof