Technical Summary: Distributional Instrumental Variable (DIV) Method

Problem Statement

The paper addresses the challenge of inferring causal effects in the presence of unmeasured confounding (hidden confounders $H$) when randomised experiments are unavailable. While the Instrumental Variable (IV) approach is a standard tool for such endogeneity, existing methods typically focus on estimating specific functionals of the causal effect, such as the mean causal effect (e.g., via Two-Stage Least Squares, 2SLS) or quantile treatment effects.

The authors argue that summarizing causal effects into single quantities (like means) discards valuable information. In many applications, understanding the entire interventional distribution of the outcome $Y$ under a $do$-intervention on treatment $X$ (denoted $P_{do(X:=x)}(Y)$) is more informative. Furthermore, traditional IV methods often fail in "under-identified" settings where the number of instruments is smaller than the number of treatment variables, or when the relationship between the instrument and treatment is weak in terms of conditional means but strong in higher-order distributional moments.

Methodology: Distributional Instrumental Variable (DIV)

The proposed Distributional Instrumental Variable (DIV) method estimates the full interventional distribution by leveraging generative modelling within a Structural Causal Model (SCM) framework.

1. Structural Causal Model and Generative Formulation

The authors assume the data is generated by an SCM where:
$$X := g(Z, \eta_X)$$
$$Y := f(X, \eta_Y)$$
Here, $Z$ is the instrument, $X$ is the treatment, $Y$ is the outcome, and $\eta_X, \eta_Y$ are noise terms correlated due to hidden confounding. Unlike classical linear IV methods, DIV allows for general nonlinear functions $g$ and $f$ and does not assume additive noise.

To capture the correlation between $\eta_X$ and $\eta_Y$, DIV models the noise terms as functions of independent standard Gaussian variables and a shared latent noise component:
$$\eta_X = h_X(\varepsilon_X, \varepsilon_H)$$
$$\eta_Y = h_Y(\varepsilon_Y, \varepsilon_H)$$
where $\varepsilon_X, \varepsilon_Y, \varepsilon_H$ are independent standard Gaussians. This yields a joint generative model for $(X, Y)$ given $Z$.

2. Estimation via Energy Score

The core of the DIV method is fitting the joint conditional distribution $P(X, Y | Z)$ using Engression (Shen and Meinshausen, 2024). The model parameters (neural networks for $g, f, h_X, h_Y$) are optimized to minimize the expected negative energy score, a strictly proper scoring rule for multivariate distributions.

The population objective is:
$$(g^*, f^*, h_X^*, h_Y^*) \in \arg \min_{f,g,h_X,h_Y} \mathbb{E} \left[ \|(X, Y) - (\hat{X}, \hat{Y})\| - \frac{1}{2}\|(\hat{X}, \hat{Y}) - (\hat{X}', \hat{Y}')\| \right]$$
where $(\hat{X}, \hat{Y})$ and $(\hat{X}', \hat{Y}')$ are independent samples generated by the model.

3. Interventional Distribution Estimation

Once the model is fitted, the interventional distribution $P_{do(X:=x)}(Y)$ is estimated via sampling. A $do$-intervention $do(X:=x)$ severs the link between $X$ and its noise $\eta_X$. The procedure involves:

1. Sampling $\varepsilon_Y$ and $\varepsilon_H$ from standard Gaussians.
2. Computing the noise $\eta_Y^* = h_Y^*(\varepsilon_Y, \varepsilon_H)$.
3. Generating the outcome $Y^* = f^*(x, \eta_Y^*)$.

From these samples, any functional of the distribution (mean, quantiles, variance) can be estimated via Monte Carlo.

Key Theoretical Contributions

The paper establishes identifiability results for the interventional distribution under three distinct model configurations:

1. General Model Class: Under assumptions of strict monotonicity in the latent disturbances and a common support condition (requiring the instrument to have a large support), the interventional distribution is uniquely identifiable from the observed data.
2. Pre-Additive Noise Models (Continuous Instrument): By restricting the outcome model to a pre-additive noise form, the authors show identifiability even when the instrument has a continuous support but does not necessarily satisfy the "large support" condition required for the general class.
3. Pre-Additive Noise Models (Discrete/Binary Instrument): Crucially, the paper demonstrates that for pre-additive noise models, the interventional distribution is identifiable even with a binary instrument and multivariate treatment (an under-identified setting where $dim(Z) < dim(X)$). This contrasts with 2SLS, which fails in such cases due to multicollinearity in the first stage. The authors show that if the instrument affects the treatment distribution in a way that is not merely a deterministic shift (i.e., the conditional distributions differ beyond a mean shift), the full distributional information suffices for identification.

The paper also proves the consistency of the DIV estimator in a simple parametric linear subclass.

Empirical Results

The authors evaluate DIV against benchmarks including 2SLS, Control Functions (CF), DeepIV, DeepGMM, HSIC-X, IVQR, and DIVE.

• Simulated Data:
  • Mean and Quantile Estimation: DIV performs competitively or superiorly in estimating interventional means and quantiles across linear, nonlinear, and pre-additive noise settings.
  • Under-Identified Cases: In scenarios with a single binary instrument and two-dimensional treatment ($dim(Z)=1, dim(X)=2$), 2SLS and other benchmarks fail to identify causal parameters. DIV successfully recovers the parameters by utilizing the full distributional information.
  • Weak Instruments: DIV remains robust when the instrument has weak mean relevance or weak mean-variance relevance, provided higher-order distributional moments carry structural information.
• Real-World Applications:
  • Economic Data (Acemoglu et al., 2001): Applied to the relationship between institutional quality and GDP. DIV recovers a positive, roughly linear causal effect consistent with the original 2SLS findings, while also providing a full distributional estimate (quantiles).
  • Single-Cell Data: Applied to gene expression prediction under unseen CRISPR perturbations. DIV demonstrates superior generalizability to unseen intervention environments and stability across different training subsets compared to other methods, suggesting it captures the underlying causal mechanism more effectively.

Significance and Claims

The paper claims that DIV offers a significant advancement in causal inference by:

1. Expanding the Estimand: Moving beyond mean or quantile effects to estimate the entire interventional distribution, providing a richer understanding of causal mechanisms and aleatoric uncertainty.
2. Overcoming Identification Limits: Solving "under-identified" problems where traditional IV methods (like 2SLS) fail, specifically by leveraging distributional information in settings with discrete instruments and multivariate treatments.
3. Flexibility: Accommodating general nonlinear relationships without restrictive additive noise assumptions.
4. Practical Utility: Providing a computationally efficient software implementation (R and Python) that allows for sampling from the estimated interventional distribution, facilitating the calculation of any distributional functional.

The authors conclude that while the method relies on estimating the joint distribution $P(X, Y | Z)$, this approach maximizes the use of information from the instruments, offering the best chance for identification in complex, confounded settings. Future work directions mentioned include epistemic uncertainty quantification and applying the methodology to distribution generalization tasks.