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General Probabilities of Causation with Causal Knowledge

This paper derives tighter theoretical bounds for multivalued probabilities of causation by incorporating causal information from covariates and mediators, demonstrating through simulations that these bounds improve upon existing nonbinary methods.

Original authors: Xin Shu, Zhen Lei, Ang Li

Published 2026-08-14

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CC BY 4.0

Original authors: Xin Shu, Zhen Lei, Ang Li

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 mystery, but the culprit is invisible. You can't see the person who pulled the trigger; you can only see the bullet holes and the people standing nearby. In the world of science, this is the challenge of figuring out "cause and effect." We often know that two things happen together—like eating ice cream and getting a sunburn—but knowing they happen together doesn't tell us if the ice cream caused the sunburn, or if a third thing (like a sunny day) caused both. Scientists use a special kind of math called "Structural Causal Models" to play detective. They try to calculate the "Probability of Causation" (PoC), which is basically asking: "What are the odds that this specific event caused this specific result?"

For a long time, detectives could only solve simple cases where things were either "on" or "off," like a light switch. But real life is rarely that simple. People take medicine in different amounts, and diseases can range from mild to severe. When things have many possible values (like a dimmer switch with 100 settings), the math gets incredibly messy, and the answers become very vague. It's like trying to guess the exact weight of a cloud; you might know it's somewhere between "nothing" and "a mountain," but that doesn't help you build a bridge. This paper steps into that messy middle ground to see if we can get a sharper, more useful answer by looking at the hidden clues in the background.

The authors of this paper, Xin Shu, Zhen Lei, and Ang Li, are tackling the problem of these "fuzzy" answers in complex, multi-valued situations. They ask a simple but powerful question: If we know more about the structure of the world—like how different variables are connected—can we narrow down our guesses? Think of it like trying to find a lost key in a dark room. If you only know the room is big, your search area is huge. But if you know the key was dropped near the door, or that it fell through a specific crack in the floor, your search area shrinks dramatically.

In this study, the researchers show that by using "causal graphs"—which are just maps showing how different factors influence each other—we can act like detectives with a better flashlight. They looked at two main types of clues: covariates (background factors like family history that might influence both the cause and the effect) and mediators (intermediate steps, like a drug changing your blood sugar before it changes your health).

The paper finds that when we incorporate these maps into our math, the "fuzzy" answers get much sharper. For example, in a simulated scenario about a new diabetes drug, the old method suggested the drug might work for anywhere between 0% and 55% of people. That's a huge range, making it hard to decide if the drug is actually good. But when the researchers added the "family history" clue to their map, the range shrank to 0% to just 3.3%. Suddenly, the drug looks much less effective than the company hoped. In another example involving blood glucose levels as a middle step, the range dropped from 0–50% down to 0–15%.

The researchers didn't just guess this would happen; they proved it mathematically and ran massive computer simulations with 100,000 different scenarios to test their theory. The results were consistent: whenever they used the extra causal information, the bounds (the range of possible answers) got tighter. This means we can rule out impossible scenarios and get a much clearer picture of what is actually causing what. While the math is complex, the takeaway is playful and profound: by understanding the hidden connections in the web of life, we can stop guessing in the dark and start seeing the truth with much greater clarity.

Technical Summary: General Probabilities of Causation with Causal Knowledge

Problem Statement

Probabilities of causation (PoCs), such as the probability of necessity (PN), probability of sufficiency (PS), and probability of necessity and sufficiency (PNS), characterize individual causal responses that are generally unobservable and require partial identification. While Tian and Pearl (2000) established theoretically sharp bounds for binary treatments and outcomes using experimental and observational data, and subsequent works (Mueller et al., 2021; Li and Pearl, 2019, 2022b) demonstrated that incorporating covariates and mediators could tighten these bounds in binary settings, a significant gap remained for nonbinary (multivalued) settings.

Recent extensions by Zhang et al. (2022), Li and Pearl (2022a, 2024), and Shu, Wang, and Li (2026) derived bounds for multivalued PoCs. However, it remained an open question whether additional causal knowledge encoded in causal graphs (specifically regarding covariates and mediators) could further tighten these multivalued bounds. The complexity of multivalued PoCs, which involve joint probabilities of multiple potential outcomes, introduces a substantially more complex counterfactual structure than the binary case, making the immediate extension of binary results non-trivial.

Methodology

This paper addresses the open question by deriving tighter bounds for multivalued PoCs through the incorporation of causal information encoded in covariates and mediators. The authors operate within the Structural Causal Model (SCM) framework.

The methodology involves:

Defining Multivalued PoCs: The paper focuses on four representative forms of multivalued PoCs introduced by Shu, Wang, and Li (2026), which are sufficient to represent any discrete PoC:
- $PNS(k)$: Probability of necessity and sufficiency for $k$ treatment levels.
- $PSub(k, p)$: Probability of substitute.
- $PRep(k, q)$: Probability of replacement.
- $PN(k, p, q)$: Probability of necessity.
Integrating Causal Graphs: The authors derive new bounds by imposing structural constraints on feasible joint counterfactual distributions based on specific causal graph structures. They consider:
- Non-descendant Covariates: Variables $Z$ that are not descendants of the treatment $X$ . When $Z$ satisfies the back-door criterion, the bounds can be computed using only observational data via the back-door adjustment formula.
- Partial Mediators: Variables $Z$ that are descendants of $X$ where $X$ also has a direct effect on the outcome $Y$ .
- Pure Mediators: Variables $Z$ where $X$ affects $Y$ exclusively through $Z$ (no direct effect).
Mathematical Derivation: For each causal structure, the authors derive improved upper and lower bounds by integrating available experimental and observational distributions with graph-based constraints. The proofs rely on Fréchet inequalities and properties of conditional independence derived from the causal graphs (e.g., cross-world conditional independences).

Key Contributions

The paper makes three primary contributions:

Extension to Multivalued Settings: The authors extend existing bounds on PoCs that incorporate causal knowledge from binary to multivalued settings. Unlike the binary case, the multivalued setting requires characterizing joint counterfactual probabilities across multiple treatment levels, introducing new technical challenges.
Comprehensive Framework: The results cover the four representative forms of multivalued PoCs ($PNS$, $PSub$, $PRep$, $PN$) introduced by Shu, Wang, and Li (2026). Since these forms can represent any discrete PoC, the proposed framework immediately applies to a broad class of multivalued PoCs.
Diverse Causal Structures: The paper derives specific bounds for multiple causal structures, including non-descendant covariates (including confounders and backdoor adjustment sets), partial mediators, and pure mediators.

Results

The paper validates the proposed theorems through toy examples and simulation studies:

Toy Examples:
- In a diabetes prevention scenario with family history as a covariate, incorporating the causal structure tightened the upper bound of $PNS(3)$ from 0.551 (using only marginal data) to 0.033.
- In a scenario with blood glucose as a pure mediator, the upper bound tightened from 0.507 to 0.151.
- These examples demonstrate that ignoring causal structure can lead to wide, less informative bounds that may mislead decision-making (e.g., overestimating drug efficacy).
Simulation Studies:
- The authors generated 100,000 sample distributions compatible with the causal diagrams for various dimensions ( $n, k \in \{3, 4, 5\}$ ).
- Non-descendant Covariates: This structure provided the most substantial tightening. For $n=k=5$ , 99.85% of sample distributions benefited from tighter bounds, with the average width of the bounds decreasing significantly compared to the baseline (Shu et al., 2026).
- Pure Mediators: This structure also yielded significant improvements (e.g., 83.66% of cases improved for $n=k=3$ ).
- Partial Mediators: The improvement was marginal and decreased as dimensionality increased. For $n=k=4$ , no bounds were tightened in 20 million draws, suggesting that while theoretically valid, the practical utility of partial mediators in high dimensions is limited compared to other structures.

Significance and Claims

The paper claims that incorporating causal graph information consistently yields tighter bounds on individual causal responses than approaches based solely on experimental and observational distributions. By extending the use of causal structure from binary to multivalued settings, the work shows when and how auxiliary variables (covariates and mediators) can exclude structurally incompatible response patterns.

The authors emphasize that while the bounds derived by Tian and Pearl are sharp given only distributional information, the inclusion of causal structure provides a mechanism to narrow these bounds further. The significance lies in providing a theoretical foundation for more precise identification of causal effects in complex, multivalued scenarios, thereby offering more informative bounds for decision-making in fields such as pharmaceutical evaluation. The paper remains modest, noting that the partial-mediator bounds, while theoretically valid, may offer limited practical improvement in high-dimensional settings.