Incremental Propensity Score Interventions

• Incremental propensity score interventions are a stochastic treatment policy that multiplies odds to adjust treatment probabilities across populations.
• They offer closed-form identifiability and enable semiparametrically efficient estimation via methods like TMLE, one‐step, and cross-fitted estimators.
• These interventions facilitate robust policy evaluation and cost-sensitive decision making without the need for strict deterministic positivity.

Incremental propensity score interventions (IPSI) are a general class of stochastic treatment policies that systematically multiply the odds of receiving treatment by a user-specified factor across the population, as opposed to setting treatment status deterministically. This construction targets causal effects under policy shifts and avoids strict positivity assumptions, enabling robust estimation even in the presence of unmeasured confounding or positivity violations. IPSI unify and extend classic binary incremental interventions, are well-defined in longitudinal and multivariate settings, and connect to cost-sensitive decisions through entropy-regularized couplings. These interventions admit closed-form characterizations of identifiability, influence functions, and estimation protocols that achieve semiparametric efficiency under nonparametric modeling, with plug-in, one-step, @@@@1@@@@, and cross-fitted estimators yielding valid inference and uniform confidence bands over intervention intensities.

1. Formal Definition and Mathematical Structure

Incremental propensity score interventions act by modifying the individual probability of receiving treatment via odds multiplication. For a binary treatment $A \in \{0,1\}$ with propensity score $\pi(x) = P(A=1|X=x)$ for covariate profile $x$, the classic IPSI defines the intervention regime as:

$$q_\delta(1|x) = \frac{\delta\;\pi(x)}{\delta\;\pi(x) + (1-\pi(x))}$$

$$q_\delta(0|x) = \frac{1-\pi(x)}{\delta\;\pi(x) + (1-\pi(x))}$$

where $\delta>0$ is the odds multiplier. This construction generalizes to discrete or multi-arm treatments by tilting the joint law via a cost-penalized information projection. The most recent framework introduces a coupling $\gamma$ on $(A', A'')$ that solves a penalized KL minimization:

$$\min_\gamma D_{KL}(\gamma \Vert \pi(\cdot|x)\otimes\nu) + \delta\, E_\gamma[c(A',A'')]$$

yielding a Boltzmann-Gibbs product:

$$\gamma^*_\delta(a',a''|x) \propto \pi(a'|x)\,\nu(a'')\,e^{-\delta c(a',a'')}$$

for user-specified target $\nu$ and cost $c$, with normalizing constant $Z(x;\delta)$ (Aguas, 14 Nov 2025). Marginalizing yields two classes of stochastic policies: source-tilted $\pi^*_\delta$ and target-tilted $\nu^*_\delta$, both of which interpolate between organic and log-odds-shifted regimes as $\delta$ varies.

2. Identification: Assumptions and Formulas

The identification of IPSI causal effects proceeds under standard assumptions:

• Consistency: $Y = Y^a$ if $A=a$
• Conditional exchangeability: $A \perp (Y^0,Y^1)|X$
• No additional positivity is required, as the stochastic shift never forces zero-support units into treatment (Bonvini et al., 2021, Kennedy, 2017)

The mean counterfactual outcome under the incremental regime is:

$$\psi(\delta) = E_X\big[q_\delta(1|X)\,\mu(1,X) + q_\delta(0|X)\,\mu(0,X)\big]$$

where $\mu(a,x) = E[Y|A=a,X=x]$. For longitudinal designs with timepoints $t=1,\ldots,T$, odds are shifted multiplicatively at each $t$ and identification via the g-formula remains valid even without global positivity (Kim et al., 2019):

$$\psi(\delta) = \mathbb{E}\left[Y \prod_{t=1}^{T} \frac{\delta A_t + (1-A_t)}{\delta \pi_t(H_t) + 1 - \pi_t(H_t)} \right]$$

Conditional incremental effects (CIE), contrasts (CICE), and derivatives (CIDE) have closed forms for all subgroups $V\subseteq X$ and for infinitesimal policy shifts (McClean et al., 2022):

$$\tau_{cie}(v;\delta) = E\left[ \frac{\delta\,\pi(X)\,\mu(1,X)+(1-\pi(X))\,\mu(0,X)}{\delta\,\pi(X)+(1-\pi(X))} \mid V=v \right]$$

No positivity is required: if $\pi(x)\in\{0,1\}$, the intervention leaves such units unchanged.

3. Estimation Algorithms and Inference Procedures

Efficient estimation of IPSI effects is achieved using doubly-robust one-step estimators, TMLE, IPW, and projection/meta-learners with cross-fitting:

• Influence function-based estimator: Evaluate the canonical EIF using plug-in or cross-fitted machine learning nuisance estimates for $\pi$ and $\mu$ (Kennedy, 2017, McClean et al., 2022).
• One-step estimator: Average EIF per sample, yielding:

  $$\widehat\psi(\delta) = n^{-1}\sum_{i=1}^{n} \varphi(Z_i; \delta, \widehat\pi, \widehat\mu)$$

• TMLE: Target $\psi(\delta)$ via adaptive fluctuation of the outcome regression.
• Projection-learner and I-DR-learner for conditional effects: Nonparametric meta-learning of pseudo-outcomes, maintaining double robustness (McClean et al., 2022).
• Uniform confidence bands: Employ multiplier (wild) bootstrap to derive simultaneous bands over a grid of $\delta$ (Kennedy, 2017).

Under $n^{-1/4}$ convergence rates of nuisance functions, estimators are $\sqrt{n}$-consistent and attain the semiparametric efficiency bound. Asymptotic variance is estimated using the sample variance of influence-function contributions.

Method	Nuisance learning	Robustness
One-step EIF	ML, cross-fitting	Doubly robust
TMLE	ML, cross-fitting	Multiply robust
I-DR-learner	ML nonparametric	Double robust

4. Extensions: Multivariate, Cost-Aware, and Sensitivity Analysis

The cost-penalized I-projection framework generalizes IPSI to:

• Multi-arm and discrete treatments: Couplings $\gamma$ control policy shifts across arbitrary action sets, with source-tilted and target-tilted marginals (Aguas, 14 Nov 2025).
• Cost-sensitive interventions: The tilt parameter $\delta$ and cost function $c(a)$ weight the transition toward cost-effective target policies. When $c(a)>0$ strictly, the policy converges to a product-of-experts law as $\delta\rightarrow\infty$.
• Sensitivity to unmeasured confounding: Rosenbaum’s $\Gamma$-selection-bias model enables partial identification via estimable bounds, sharp under single-time and extensible to time-varying settings via marginal sensitivity models (Shen et al., 25 Jan 2026). Feasible estimators for bounds (doubly robust, sample-split) and convexity properties are established.

5. Empirical Applications and Interpretations

IPSI have been applied in diverse substantive domains:

• ICU admission and mortality: CIE and CIDE curves reveal no substantial effect heterogeneity or mortality reduction under plausible policy shifts; uniform bands cover the status quo (McClean et al., 2022).
• Recidivism and homelessness: Large reductions in homelessness odds correspond to statistically significant decreases in rearrest rates, with smaller odds-shifts yielding less impact (Jacobs et al., 2023).
• Incarceration-marriage: Doubling incarceration odds marginally reduces marriage rates; uniform tests over $\delta$ detect statistically significant effects, even without positivity (Kennedy, 2017).
• Clinical trials with dropout: Incremental IPW and one-step estimators avoid exploding-variance and remain reliable where deterministic effects fail due to positivity or dimensionality (Kim et al., 2019).

Simulations consistently show that efficient, cross-fitted estimators outperform plug-in and parametric estimators, particularly under model misspecification and rare treatment arms.

6. Practical Considerations and Limitations

• Choice of $\delta$ values: The intervention intensity ($\delta$) should be chosen to reflect plausible, policy-relevant shifts. Interpretation is direct: $\psi(\delta)$ is the average outcome if the population’s odds were uniformly scaled.
• Positivity and support: Extreme $\delta$ may cause effective propensity violation—screen for unstable weights.
• Confounding: No unmeasured confounding (exchangeability) is essential; sensitivity analysis with bound estimators is recommended in practical deployment.
• Tuning and cross-validation: Cross-fitting or sample-splitting is critical for avoiding overfit, especially with flexible ML nuisance estimation.
• Estimator choice: One-step and TMLE estimators provide valid inference under mild conditions and can incorporate ensemble learning methods.
• Interpretability: The IPSI effect is not an average treatment effect (ATE); it describes a continuum of outcomes as a function of the policy shift.

7. Connections, Generalizations, and Future Directions

The IPSI paradigm resides at the intersection of stochastic policy evaluation, semiparametric inference, and experimental design:

• Entropy-regularized coupling: The Boltzmann-Gibbs structure links causal effect estimation to optimal transport and cost-sensitive decision analysis (Aguas, 14 Nov 2025).
• Dose-response curve: The $\delta$-curve captures a spectrum from observed policy ($\delta=1$) through increased intervention intensity.
• Extensions to continuous exposures, instrumental variables, network-interference, and optimal policy design are noted as active research areas (Bonvini et al., 2021).
• Sensitivity bounds, identification in data integration, transportability, and efficient estimation in high-dimensional or longitudinal settings remain major foci (Wen et al., 26 Sep 2025, Shen et al., 25 Jan 2026).

Incremental propensity score interventions thus offer a comprehensive, robust framework for causal effect estimation, policy evaluation, and actionable inference under both observational and experimental regimes.