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On the Recoverability of Causal Relations from Bulk Gene Expression Data

This paper demonstrates that recovering causal relations from aggregated bulk gene expression data is theoretically possible only under strict linear aggregation and affine structural equation assumptions, which are empirically unsupported by real-world datasets, thereby cautioning against such causal inference without strong additional constraints.

Original authors: Gongxu Luo, Boyang Sun, Kun Zhang

Published 2026-06-02

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CC BY 4.0

Original authors: Gongxu Luo, Boyang Sun, Kun Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to understand how a complex machine works by looking at a single, blurry photograph of the whole thing, rather than seeing the individual gears and springs moving inside. This is essentially the challenge scientists face when studying bulk gene expression data.

Here is a simple breakdown of what this paper discovered, using everyday analogies.

The Problem: The "Smoothie" Effect

In biology, scientists often want to know how genes talk to each other. Does Gene A turn Gene B on? Does Gene C stop Gene D?

To find this out, researchers usually take a sample of tissue (like a drop of blood) and measure the RNA inside. However, a drop of blood contains millions of individual cells. Traditional methods don't look at each cell one by one; instead, they blend all the cells together into a "smoothie" (this is called bulk expression).

The Analogy: Imagine you have a room full of people (cells) shouting different instructions. If you record the room with a single microphone (bulk data), you only hear a loud, mixed-up roar. You lose the specific details of who said what to whom.
The Hope: For years, scientists have tried to use computer algorithms to figure out the original "instructions" (causal relationships) just by listening to that mixed-up roar. They hoped that if they analyzed the noise carefully, they could reconstruct the original conversation.

The Big Discovery: The Smoothie Hides the Truth

This paper asks a fundamental question: Is it actually possible to reconstruct the original conversation just from the mixed-up roar?

The authors say: Generally, no.

They proved mathematically that when you blend cells together, you often destroy the very clues needed to figure out cause-and-effect. The "smoothie" changes the rules of the game.

The Two Rules for Recovery

The researchers found that you can only recover the original causal relationships if two very strict conditions are met. Think of these as the "Golden Rules" for un-blending the smoothie:

The "Straight Line" Rule (Functional Consistency):
The way genes affect each other must be perfectly straight and simple (linear).
- Analogy: Imagine Gene A adds exactly 10 units of energy to Gene B, no matter what. If Gene A adds 10, and Gene B adds 10 to Gene C, the math is simple.
- The Reality: In real life, genes are messy. Sometimes Gene A adds 10, sometimes 50, sometimes it stops working entirely. It's a curve, not a straight line. The paper shows that if the relationship is curved (non-linear), blending the cells destroys the pattern forever.
The "Perfect Blend" Rule (Conditional Independence):
The way the cells are mixed together must be a simple average or sum.
- Analogy: If you mix red and blue paint to get purple, you can mathematically reverse it only if you mixed them in a perfectly predictable way. If the mixing process is weird (like taking the "middle" color or the "brightest" color), you can't reverse it.
- The Reality: The paper proves that only simple averaging works. If the biological system is complex, the math breaks.

The "Real World" Test

The authors didn't just do math on paper; they tested this with real data.

They looked at synthetic data (computer-generated scenarios where they knew the answer). As predicted, when the genes acted in simple, straight-line ways, the math worked. When they acted in complex, curved ways, the math failed completely.
They looked at real human gene data (both from bulk samples and single cells). They found that genes do not act in simple, straight lines. The relationships are complex and curved.

The Conclusion

Because real genes behave in complex, non-linear ways, and because bulk data blends cells together, you cannot reliably figure out the true cause-and-effect relationships between genes just by looking at bulk data.

The Metaphor: Trying to find the specific conversation between two people in a crowded room just by listening to the average noise level is impossible. The noise (aggregation) has erased the specific details.

What This Means for Science

The paper warns scientists: Don't trust causal maps made from bulk data unless you have strong extra proof.

If a study claims "Gene A causes Gene B" based only on bulk data, that claim might be an illusion created by the blending process, not a real biological fact. To get the truth, scientists likely need to look at individual cells (single-cell data) or make very strong, specific assumptions that the paper suggests are rarely true in nature.

In short: The "smoothie" of bulk data is too messy to let us see the individual ingredients' recipes. We need to look at the ingredients separately to understand the recipe.

Technical Summary: On the Recoverability of Causal Relations from Bulk Gene Expression Data

Problem Statement
While single-cell assays are increasingly available, bulk gene expression profiling remains a prevalent, cost-effective, and less noisy method for large-scale biological studies. Bulk data represents an aggregation of RNA pooled across many cells within a sample. A significant body of computational work attempts to infer causal gene regulatory networks directly from these aggregated bulk measurements, typically treating them as direct observations of a single homogeneous causal system. However, aggregation is a lossy, non-invertible coarsening of the underlying cellular system. This paper addresses a fundamental, underexplored question: Under what conditions can causal relations among genes be reliably recovered from aggregated bulk expression data, and when does the aggregation process distort the identifying information (functional form, statistical dependence, and conditional independence) necessary for causal discovery?

Methodology and Theoretical Framework
The authors formalize the problem of causal recoverability under aggregation through two complementary notions of consistency between the underlying latent causal model and the observed aggregated data:

Functional-Form Consistency: This examines whether the aggregated variables admit a Structural Causal Model (SCM) of the same functional family as the underlying model. The authors analyze two common functional causal model families: the Additive Noise Model (ANM, $Y = f(X) + \epsilon_Y$ ) and the Post-Nonlinear (PNL) model ( $Y = f_2(f_1(X) + \epsilon_Y)$ ).
Conditional-Independence (CI) Consistency: This investigates whether the conditional independence relations among latent variables (e.g., $X \perp\!\!\perp Z | Y$ ) are preserved in the aggregated variables ( $\bar{X} \perp\!\!\perp \bar{Z} | \bar{Y}$ ).

The study derives necessary and sufficient conditions for these consistencies to hold. The analysis distinguishes between linear aggregations (e.g., sum, mean) and nonlinear aggregations (e.g., median, max).

Key Theoretical Results
The paper establishes that causal relations are recoverable from aggregated data only under highly restrictive conditions:

Functional-Form Recovery: For functional-form consistency to be preserved under linear aggregation (e.g., sum/mean), the underlying structural equations must be affine (i.e., $f(X) = \alpha X + \beta$ $f (X) = α X + β$ ). Specifically:
- For ANM, the function $f$ must be affine.
- For PNL, both $f_1$ and $f_2$ must be affine.
- If the aggregation is nonlinear (e.g., median), functional-form consistency is generally not preserved even for affine relations.
Conditional-Independence Recovery: For the conditional independence structure to be preserved (specifically for chain and fork structures where $X \perp\!\!\perp Z | Y$ ), the underlying relationships must be linear. In nonlinear systems, conditioning on the aggregated variable $\bar{Y}$ fails to block the path between $\bar{X}$ and $\bar{Z}$ , leading to spurious dependencies that violate the true latent structure.

Empirical Evaluation
The authors validate these theoretical findings through two phases:

Synthetic Data: Controlled simulations confirm that causal discovery methods (both functional-based and constraint-based) maintain high precision only when the ground truth involves linear aggregation combined with affine structural equations. When nonlinear functions or nonlinear aggregations are introduced, the precision of recovering causal directions and conditional independencies drops significantly.
Real-World Data: The authors analyze four bulk gene expression datasets (two microarray, two RNA-seq) and four single-cell datasets. They employ statistical diagnostics to assess the linearity of pairwise gene regulatory functions:
- Inverse Effective Degrees of Freedom (1/edf): Derived from Generalized Additive Models (GAMs) to measure the flexibility required to fit the data.
- Signal-Weighted Linearity Score: Measures the deviation of estimated pairwise functions from their best affine approximations.
The results indicate that in both bulk and single-cell data, the estimated pairwise regulatory functions deviate significantly from linearity. While bulk aggregation induces some smoothing that increases apparent linearity compared to single-cell data, the assumption of affine relationships required for theoretical recoverability is not supported by the empirical evidence.

Significance and Claims
The paper concludes that recovering causal relations from aggregated bulk expression data without strong additional assumptions is generally not feasible. The theoretical conditions required for recoverability (linear aggregation coupled with affine structural equations) are rarely met in real transcriptomic data, where gene-gene interactions are predominantly nonlinear.

The authors caution against interpreting causal claims derived solely from bulk expression profiles as direct reflections of underlying cell-level causal mechanisms. They argue that the field must recognize the fundamental limits of bulk data for causal discovery and that reliable recovery likely requires auxiliary information, such as perturbation data, time-series measurements, or single-cell references, rather than relying on bulk data alone. The work provides a principled boundary for what can and cannot be inferred from aggregated bulk cohorts.