Fundamental Problem of Causal Inference

Definition

The problem that, for the same individual, the outcomes under treatment (W=1) and control (W=0) cannot be observed simultaneously

The core problem of causal inference, formalized by Holland (1986):

“The fundamental problem of causal inference is that we can observe at most one of the potential outcomes for each unit.”

$$\text{ITE}_i = Y_i(1) - Y_i(0)$$

In the equation above, only one of $Y_i(1)$ and $Y_i(0)$ can be observed.

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Intuitive Understanding

Example: Drug Effect

When drug A is administered to patient Alice, only the outcome under that condition can be observed:

Patient	Drug A outcome (observed)	No-drug outcome (counterfactual)
Alice	Y=recovered	? (unobservable)

Counterfactual outcome:

• “What would have happened if Alice had not received the drug?”
• The answer to this question cannot be observed directly

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Missing Data Perspective

Causal inference can be viewed as a missing data problem:

Unit	W	Y(1)	Y(0)	ITE
1	1	5	?	?
2	0	?	3	?
3	1	7	?	?
4	0	?	4	?

• Treatment group: $Y(1)$ observed, $Y(0)$ missing
• Control group: $Y(0)$ observed, $Y(1)$ missing

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Solution Strategies

1. Group-level Comparison (ATE estimation)

Estimate the average treatment effect (ATE) instead of the individual-level ITE:

$$\text{ATE} = E[Y(1)] - E[Y(0)]$$

Estimable from observational data under assumptions.

2. Leveraging Key Assumptions

Identification is possible when the three key assumptions hold:

• SUTVA
• Ignorability
• Positivity

3. Matching / Weighting

Infer counterfactual outcomes by comparing similar individuals:

• Propensity Score Matching: match individuals with similar propensities
• IPW: balance group distributions with weights

4. Model-based Estimation

Estimate potential outcomes with a predictive model:

• T-Learner: an outcome model for each treatment group
• S-Learner: estimate the treatment effect with a single unified model

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RCT vs Observational Study

RCT (Randomized Controlled Trial)

Random assignment automatically satisfies Ignorability:

$$W \perp\!\!\!\perp (Y(0), Y(1))$$

• The ATE can be estimated by comparing the means of the treatment/control groups
• “Gold standard” for causal inference

Observational Study

No random assignment → additional assumptions required:

• Conditional ignorability: $W \perp\!\!\!\perp (Y(0), Y(1)) \mid X$
• Covariate adjustment required

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Why Does It Matter?

1. Prediction ≠ Causation

   • Prediction: “Given X, what is Y?”
   • Causation: “If we change X, how does Y change?”

2. The effect of intervention

   • Essential for decision-making in policy, treatment, marketing, etc.

3. Counterfactual thinking

   • The ability to answer “what if we had done it differently?”

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Related Concepts

• Potential Outcomes - Rubin Causal Model
• Causal Assumptions Overview - integrated overview of the assumptions
• Counterfactual Reasoning - counterfactual reasoning
• Selection Bias - bias due to differences between treatment/control groups
• ITE - Individual Treatment Effect

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References

• Holland, P. W. (1986). Statistics and Causal Inference. JASA
• yaoSurveyCausalInference2021 - Section 2.5
• Potential Outcomes - Rubin (1974) framework