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Comparing Two Proxy Methods for Causal Identification

This paper contrasts two major proxy variable approaches for causal identification—bridge equation methods and array decomposition methods—by analyzing their underlying model restrictions and assumptions to clarify the scope of applicability for each.

Original authors: Helen Guo, Elizabeth L. Ogburn, Ilya Shpitser

Published 2026-05-12

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Helen Guo, Elizabeth L. Ogburn, Ilya Shpitser

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a crime, but there is a crucial witness who is hiding in the shadows. This witness, let's call him "The Unseen Cause" (or U), is the reason both the suspect (the Treatment) and the victim (the Outcome) are acting the way they are. Because you can't see or talk to this witness directly, you can't easily prove who did what to whom. This is the classic problem of unmeasured confounding in causal inference.

However, you do have two other witnesses who were in the room: Proxy 1 (a friend of the suspect) and Proxy 2 (a friend of the victim). Neither of them saw the crime directly, but they both know the hidden witness.

This paper, written by Helen Guo, Elizabeth Ogburn, and Ilya Shpitser, compares two different detective strategies for using these two proxies to figure out what the hidden witness would have said, and ultimately, to solve the crime (identify the causal effect).

The Two Detective Strategies

The paper contrasts two major ways of solving this puzzle: The Bridge Equation Method and The Array Decomposition Method.

1. The Bridge Equation Method (The "Direct Translator")

Think of this as trying to translate a secret message directly.

This approach, introduced by Miao et al. (2018), tries to build a direct "bridge" between what you observe (the proxies) and the answer you need (the causal effect).

How it works: It assumes there is a mathematical "bridge" (an integral equation) that connects the behavior of the proxies to the hidden cause. If you can solve this equation, you can skip the middleman and calculate the answer directly.
The Catch: This method is very specific. It requires that your proxies are "complete" enough to cover all the secrets of the hidden witness. Imagine trying to translate a book; you need a dictionary that covers every single word. If your dictionary (the proxy) is missing even a few words, the translation fails.
What it gets you: It gives you the final answer (the causal effect) directly. It does not necessarily tell you what the hidden witness looks like or what his specific habits are; it just tells you the result of the crime.

2. The Array Decomposition Method (The "Puzzle Solver")

Think of this as taking apart a complex 3D puzzle to see how the pieces fit together.

This approach, rooted in the work of Kuroki, Pearl, and Kruskal, takes a different route. Instead of trying to jump straight to the answer, it tries to reconstruct the entire story of the hidden witness and the proxies.

How it works: It treats the data as a giant, multi-dimensional puzzle (a "tensor" or "array"). By looking at how the proxies and the outcome move together, it uses a mathematical technique called "eigendecomposition" (think of it as finding the unique patterns or "frequencies" in the data) to separate the hidden witness from the noise.
The Catch: It requires that the puzzle pieces are distinct enough to be separated. If the hidden witness has too many different "personalities" (categories) compared to the number of clues your proxies give you, the puzzle pieces might get mixed up, and you can't tell them apart.
What it gets you: It recovers the full story. It figures out exactly who the hidden witness is, what his habits are, and how he interacts with everyone. Once you have the full story, you can calculate the causal effect.

The Big Showdown: How Do They Compare?

The authors put these two methods side-by-side to see when one works and the other doesn't.

Different Rules for Different Games: The paper finds that these two methods rely on different "rules of the game" (conditional independence assumptions). They are non-nested, meaning one is not simply a special case of the other.
- Sometimes, the Bridge Method works because the math allows a direct translation, even if the puzzle pieces aren't perfectly distinct.
- Sometimes, the Array Method works because the puzzle pieces are distinct enough to be separated, even if a direct translation equation doesn't exist.
The Overlap: There is a middle ground where both methods work. This happens when the proxies are very strong and the hidden cause is simple enough that both the "translation" and the "puzzle" approaches succeed.
The Trade-off:
- The Bridge Method is like a shortcut: it gets you the answer faster but requires a very specific, strong connection between the proxies and the hidden cause.
- The Array Method is like a thorough investigation: it takes more time and requires more distinct clues to reconstruct the whole picture, but it gives you a deeper understanding of the hidden cause itself.

The "Discrete" vs. "Continuous" World

The paper also explains that these methods work slightly differently depending on the type of data:

Discrete Data (Categories): Imagine the hidden witness has a finite number of "moods" (Happy, Sad, Angry). In this world, the math is like counting blocks. The "Array Method" is very clear here: you just need enough different blocks to separate the moods.
Continuous Data (Smooth Scales): Imagine the hidden witness has a mood that can be any number on a scale from 1 to 100. Here, the math becomes more like solving a complex wave equation. The paper shows that the "Bridge Method" relies on "completeness" (having enough information to cover the whole wave), while the "Array Method" relies on "invertibility" (being able to reverse the wave to find the source).

The Bottom Line

The paper doesn't tell you which method is "better" in every situation. Instead, it acts as a guidebook for detectives. It clarifies that:

They are different tools: You can't just swap them out; they require different conditions to work.
They have different strengths: One is great for getting a direct answer if the math lines up; the other is great for reconstructing the hidden reality if the data is rich enough.
Confusion is common: Even experts in the field sometimes mix these up. This paper aims to clear the air by showing exactly where the assumptions of each method overlap and where they diverge.

In short, if you are trying to find a causal effect with hidden variables, you need to check your "detective kit." Do you have a strong bridge to cross, or do you have enough puzzle pieces to build the picture? The answer determines which method you should use.

Technical Summary: Comparing Two Proxy Methods for Causal Identification

Problem Statement
The identification of causal effects in the presence of unmeasured confounding remains a fundamental challenge in causal inference. While standard adjustment formulas (e.g., back-door adjustment) require observing all common causes, proxy variable methods have emerged as a powerful solution by leveraging variables associated with latent confounders. Within this domain, two distinct methodological frameworks have developed for nonparametric identification: the bridge equation approach (Miao et al., 2018) and the array decomposition approach (Kuroki and Pearl, 2014; Allman et al., 2015). Although both methods utilize proxies to address hidden confounding, they rely on different conditional independence structures and mathematical mechanisms. A precise comparison of their underlying assumptions, scope of applicability, and relationship to one another has been lacking, leading to active discussion even among leading contributors to the field.

Methodology and Frameworks
The paper synthesizes these two frameworks by placing them on common footing under shared conditional independences, utilizing both discrete matrix representations and continuous operator-theoretic analogues to elucidate their technical requirements.

Bridge Equation Approach:
- Mechanism: This method recovers the counterfactual distribution directly by solving integral equations (bridge equations). It re-expresses the standard confounding adjustment formula using observed proxies ( $Z$ and $W$ ) for an unmeasured confounder ( $U$ ) without identifying the full joint law of latent and observed variables.
- Key Assumptions:
  - Conditional Independence: $W \perp\!\!\perp Z, A \mid U$ and $Z \perp\!\!\perp Y \mid U, A$ .
  - Completeness (Assumption 6): A generalized linear independence condition requiring that $E[g(U) \mid Z, A] = 0 \implies g(U) = 0$ . In discrete settings, this implies the conditional distribution matrix $P_{U|Z,A}$ is right-invertible.
  - Solution to Bridge Equation (Assumption 7): Requires the existence of a function $h$ such that $f_{Y|Z,A} = \int h(w,a,y) f_{W|Z,A} dw$ . In discrete settings, this corresponds to the existence of a matrix $H$ satisfying $P_{Y|Z,A} = H P_{W|Z,A}$ .
- Outcome: Identifies the interventional distribution $f_{Y(a)}(y)$ but generally does not identify functionals dependent on $U$ (e.g., $f_{Y|U,A}$ or $f_U$ ).
Array Decomposition Approach:
- Mechanism: This method recovers the full joint law of latent and observed variables (up to latent category labels) via eigendecomposition or tensor decomposition, from which causal effects are derived. It is rooted in Kruskal's (1977) uniqueness theorem for CANDECOMP/PARAFAC (CP) decomposition.
- Key Assumptions:
  - Conditional Independence: $W \perp\!\!\perp A \mid U$ and $W, Z, Y$ are mutually independent conditional on $U, A$ .
  - Invertibility/Rank Conditions: Requires invertibility (or left-invertibility) of matrices $P_{Z|U,A}$ and $P_{W|U}$ .
  - Kruskal Rank Condition: For discrete models, the uniqueness of the decomposition relies on the inequality $2R + 2 \leq k_A + k_B + k_C$ , where $R$ is the rank (number of latent states) and $k$ denotes the Kruskal rank of the factor matrices.
  - Non-collinearity: Conditional distributions of the outcome given the latent variable must differ across latent states.
- Outcome: Identifies the full joint law $f_{A,Y,U,W,Z}$ up to a permutation of latent labels, allowing for the derivation of $f_{Y(a)}(y)$ .

Key Results and Comparisons
The paper provides a rigorous comparison of these approaches under the shared conditional independence structure depicted in Figure 3 (where $Z$ and $W$ are proxies for unmeasured confounder $U$ ).

Discrete Models:
- The assumptions of Kuroki and Pearl (2014) (Set 1) are stronger than those of Miao et al. (2018). Specifically, the invertibility of $P_{Z|U,A}$ and $P_{W|U}$ in the array decomposition approach implies the existence of the left inverse $P_{W|Z,A}$ required for the bridge equation approach. Thus, the Kuroki-Pearl model is contained within the Miao et al. model.
- However, when comparing the bridge equation approach to the generalized Kruskal conditions (Set 2), the models are non-nested. The bridge equation assumptions do not imply the Kruskal rank inequality ( $2R + 2 \leq k_A + k_B + k_C$ ), nor does the converse hold. This indicates that each framework covers distinct regions of the model space.
Continuous Models:
- The paper extends the comparison to continuous settings, referencing Hu and Schennach (2008) and Deaner (2023) for the array decomposition approach.
- The authors conjecture that the two models remain non-nested in the continuous case, similar to the discrete findings. They provide an example (a linear Gaussian model) where the assumptions of both approaches are satisfied, demonstrating an overlap in applicability.
Estimation:
- The paper briefly notes that bridge equation methods have established efficient influence functions and semiparametric efficiency results under specific parametric assumptions (e.g., location-scale families).
- In contrast, array decomposition methods often rely on sieve methods for approximation, and while semiparametric theory exists for hidden treatment proxies, it remains a nascent area for hidden confounding.

Significance and Contributions
The primary contribution of this work is the clarification of the relationship between two foundational proxy-based identification strategies. By synthesizing the literature, the authors:

Clarify Scope: Demonstrate that the two approaches are not simply alternative implementations of the same logic but rely on distinct, often non-nested, assumptions.
Unify Notation: Place both frameworks on a common footing using matrix representations for discrete cases and operator theory for continuous cases, making the technical subtleties more transparent.
Guide Application: Provide practitioners with a clearer understanding of which method is appropriate for specific data structures. For instance, if the full joint law is required (e.g., for recovering latent labels or analyzing unobserved exposures), the array decomposition approach is necessary. If only the causal effect is needed and the rank conditions for tensor decomposition are not met, the bridge equation approach may still be viable.

The paper concludes that while both methods are powerful solutions to hidden confounding, careful exposition of their underlying assumptions is essential for methodological development and applied use. Future work directions identified include extending proximal methods to more complex, generally non-identifiable scenarios and establishing general semiparametric theory for estimators derived from these approaches.