Counterfactual Reasoning

Counterfactual Reasoning

Answer "What would have happened if...?" questions by computing counterfactuals from a structural causal model. This is the highest rung of causal reasoning, enabling attribution, regret analysis, and moral/legal responsibility determination.

When to Use

• Questions about what would have happened under different circumstances
• Attribution: "Did X cause Y in this specific case?"
• Responsibility: "Who/what is responsible for this outcome?"
• Regret analysis: "Should we have done differently?"
• Legal causation: "Would the harm have occurred anyway?"
• Individual treatment effects: "Would this person have responded to treatment?"
• Explanation: "Why did this happen?"

Inputs

The Counterfactual Framework

Understanding Counterfactuals

A counterfactual asks: "Given what actually happened, what would have happened if something had been different?"

Notation:

• Y_x: The value Y would have taken had X been set to x
• P(Y_x = y | X = x', Y = y'): Probability Y would have been y under intervention x, given we observed x' and y'

Key distinction from intervention:

• Intervention (Rung 2): P(Y | do(X=x)) - What happens on average if we set X
• Counterfactual (Rung 3): P(Y_x | X=x', Y=y') - What would have happened for this specific unit

The Three-Step Counterfactual Algorithm

Step 1: Abduction (Update Background)

Use the observed evidence to infer the values of exogenous variables U.

Process:

• Start with prior distribution over U
• Condition on what was observed (X=x', Y=y', other evidence)
• Obtain posterior distribution P(U | evidence)

Intuition: Figure out what "kind of unit" this is based on what we observed.

Step 2: Action (Intervene)

Modify the structural equations to implement the hypothetical intervention.

Process:

• Replace the equation for X with X := x (the counterfactual value)
• Keep all other structural equations unchanged
• Keep the U values from Step 1

Intuition: In an alternate world, change only what the counterfactual specifies.

Step 3: Prediction (Compute)

Use the modified model with the abduced U values to compute the counterfactual outcome.

Process:

• Propagate the intervention through the causal mechanisms
• Compute Y_x given the U values from abduction
• Report P(Y_x | evidence)

Intuition: See what Y would have been in this alternate world.

Structural Causal Model Requirements

Components Needed

Structural equations: Y = f(X, U_Y)

• Specify how each variable is determined by its causes
• U represents unobserved factors

Exogenous distributions: P(U)

• Prior distribution over background factors

Graph structure: DAG implied by equations

• Which variables cause which

Level of Knowledge Required

Workflow

Step 1: Gather and Review Inputs

Collect all relevant information:

• Review the provided data and context
• Identify key parameters and constraints
• Clarify any ambiguities or missing information
• Establish success criteria

Step 2: Analyze the Situation

Perform systematic analysis:

• Identify patterns and relationships
• Evaluate against established frameworks
• Consider multiple perspectives
• Document key findings

Step 3: Generate Recommendations

Create actionable outputs:

• Synthesize insights from analysis
• Prioritize recommendations by impact
• Ensure recommendations are specific and measurable
• Consider implementation feasibility

Output Format

## Counterfactual Analysis: [Question]

### The Question
**Counterfactual query:** [Precise statement of what's being asked]
**In notation:** P(Y_x | observed evidence)

### Observed Facts
| Variable | Observed Value | Meaning |
|----------|----------------|---------|
| [Var 1] | [Value] | [Interpretation] |
| [Var 2] | [Value] | [Interpretation] |

### Causal Model

**Structural Equations:**

[Equations showing causal mechanisms]

**Graph:**

[DAG representation]

### Step 1: Abduction

**Prior beliefs about U:**
[What we initially assumed about background factors]

**Conditioning on evidence:**
[How the observed facts update our beliefs]

**Posterior inference:**
[What we now believe about U given the evidence]

### Step 2: Action

**Original equation for [treatment variable]:**
[Original structural equation]

**Modified equation (intervention):**
[Treatment variable] := [counterfactual value]

**Other equations unchanged:**
[List unchanged equations]

### Step 3: Prediction

**Computing the counterfactual outcome:**
[Show the propagation through the model]

**Result:**
[The counterfactual answer]

### Interpretation

**In plain language:**
[What this means for the specific question]

**Confidence and limitations:**
[What assumptions this depends on, sensitivity]

**Practical implications:**
[What follows from this analysis]

Common Counterfactual Questions

Causes of Effects (Attribution)

Question: "Did X cause Y in this case?" Approach: Compare factual Y with counterfactual Y_x' (what Y would have been without X)

Probability of Causation:

• PN (Probability of Necessity): P(Y_x' = 0 | X = 1, Y = 1)
  • "Would Y not have happened without X?"
• PS (Probability of Sufficiency): P(Y_x = 1 | X = 0, Y = 0)
  • "Would X have caused Y if X had occurred?"
• PNS: P(Y_x = 1, Y_x' = 0)
  • "Would X cause Y and would Y not occur without X?"

Treatment on the Treated

Question: "What was the effect of treatment on those who received it?" Formula: E[Y_1 - Y_0 | X = 1] Distinction: Different from ATE (average treatment effect on everyone)

Regret Analysis

Question: "Should we have made a different decision?" Approach: Compare actual outcome with counterfactual outcome under alternative decision

But-For Causation (Legal)

Question: "But for the defendant's action, would the harm have occurred?" Legal standard: If P(Y_x' = harm) is low, then X was a but-for cause

Example Types

Binary Example

Setup:

• X = 1 (patient took drug), Y = 1 (patient died)
• Model: Y = X*U + (1-X)*V, where U = effect if treated, V = effect if untreated

Question: "Would the patient have survived if she hadn't taken the drug?"

Abduction: Given Y=1 and X=1, we know U=1 (she died with treatment). V is unknown but we have prior beliefs.

Action: Set X := 0

Prediction: Y_0 = 0*U + (1-0)*V = V. Result depends on V distribution.

Continuous Example

Setup:

• X = years of education, Y = income, U = ability
• Model: Y = 2X + 3U + noise

Question: "What would this person's income have been with 2 more years of education?"

Abduction: From observed (X, Y), infer U

Action: Set X := X + 2

Prediction: Y' = 2(X+2) + 3U + noise = Y + 4

Constraints and Limitations

What Counterfactuals Require

• Structural model (not just causal diagram)
• Assumptions about functional forms or bounds
• Willingness to reason about unobservable scenarios

What Counterfactuals Cannot Do

• Prove what definitely happened (they give probabilities)
• Be computed from data alone without model assumptions
• Be verified empirically (the counterfactual didn't happen)

Sensitivity to Assumptions

• Counterfactual conclusions depend on:
  • Correctness of causal structure
  • Assumed functional forms
  • Distribution of unmeasured factors
• Always report sensitivity to key assumptions

Error Handling

Outputs

Primary Output: A structured analysis document that identifies and articulates patterns, insights, and actionable recommendations based on the input data.

Format:

## Analysis: [Topic]

### Key Findings
- [Finding 1]
- [Finding 2]
- [Finding 3]

### Recommendations
1. [Action 1]
2. [Action 2]
3. [Action 3]

Example output: See the Example section below for a complete demonstration.

Constraints

• Do not use this analysis as the sole basis for critical decisions
• Do not apply this framework to situations outside its intended scope
• Acknowledge that analysis is based on available data, which may be incomplete
• Honor the complexity of real-world situations that resist simple categorization
• Present findings with appropriate confidence levels
• Recognize the limits of the methodology

Example

Input:

• Question: "A patient took a new drug and died. Would she have died anyway without the drug?"
• Observed: Patient took drug (X=1), patient died (Y=1), patient was high-risk (Z=1)
• Model: We know the drug helps 80% of high-risk patients but harms 10%

Output:

Counterfactual Analysis: Drug and Patient Death

The Question

Counterfactual query: Would this patient have survived if she had not taken the drug? In notation: P(Y_0 = 0 | X = 1, Y = 1, Z = 1)

Observed Facts

Causal Model

Structural Equations:

Z = U_Z (exogenous risk status)
X = treatment decision (observed)
Y = f(X, Z, U_Y)

Where U_Y encodes individual response type:
- "Helped by drug": Y_1=0, Y_0=1 (survives with drug, dies without)
- "Harmed by drug": Y_1=1, Y_0=0 (dies with drug, survives without)
- "Always dies": Y_1=1, Y_0=1 (dies regardless)
- "Never dies": Y_1=0, Y_0=0 (survives regardless)

Population distribution (high-risk patients):

• P(Helped) = 0.80
• P(Harmed) = 0.10
• P(Always dies) = 0.08
• P(Never dies) = 0.02

Graph:

Z ---> Y
       ^
       |
X -----+

Step 1: Abduction

Prior beliefs about U_Y: Among high-risk patients: 80% helped, 10% harmed, 8% always-dies, 2% never-dies

Conditioning on evidence: We observed X=1, Y=1 (took drug and died).

Only two response types are consistent with Y=1 when X=1:

• "Harmed by drug" (Y_1=1): Possible
• "Always dies" (Y_1=1): Possible

We can rule out:

• "Helped by drug" (Y_1=0): Inconsistent with death
• "Never dies" (Y_1=0): Inconsistent with death

Posterior inference: P(Harmed | X=1, Y=1, Z=1) = 0.10 / (0.10 + 0.08) = 0.556 P(Always dies | X=1, Y=1, Z=1) = 0.08 / (0.10 + 0.08) = 0.444

Step 2: Action

Original treatment: X = 1 (patient took drug)

Counterfactual intervention: X := 0 (suppose patient had NOT taken drug)

Structural equations under intervention: Y_0 = f(X=0, Z=1, U_Y)

Step 3: Prediction

Computing Y_0 for each response type:

• If "Harmed by drug": Y_0 = 0 (would have survived)
• If "Always dies": Y_0 = 1 (would have died anyway)

Probability of survival without drug: P(Y_0 = 0 | X=1, Y=1, Z=1) = P(Harmed | evidence) × P(Y_0=0 | Harmed)

• P(Always dies | evidence) × P(Y_0=0 | Always dies) = 0.556 × 1 + 0.444 × 0 = 0.556 (about 56%)

Interpretation

In plain language: Given that this high-risk patient took the drug and died, there is approximately a 56% probability that she would have survived had she not taken the drug. Conversely, there is a 44% probability she would have died regardless.

This means:

• The drug more likely harmed this patient than not
• But we cannot be certain - she might have been a "doomed" patient
• The probability that the drug caused her death (Probability of Necessity) is 56%

Confidence and limitations:

• This analysis depends critically on the population response type distribution
• If the drug actually harmed more than 10%, the probability of causation increases
• We assumed independence between risk status and response type
• Real clinical decisions require additional factors

Practical implications:

• For legal purposes: More probable than not (>50%) that the drug caused death
• For clinical learning: This case suggests possible harm, warrants investigation
• For regulatory purposes: Individual case doesn't override population benefit (80% helped)

Important caveat: While the drug more likely than not caused this patient's death, the drug still benefits the high-risk population overall. Policy decisions should be based on population effects, not individual counterfactuals.

"The patient died after taking the drug. Would she have died anyway? This is a counterfactual question. It asks about a world that did not happen, and yes, we can answer it - with 56% probability, she would have survived without the drug."

Integration

This skill is part of the Judea Pearl expert persona. It represents the highest rung of causal reasoning. It pairs with:

• causal-diagram-construction which provides the structural foundation
• ladder-classification which identifies when Rung 3 reasoning is needed
• confounding-diagnosis which is prerequisite for identification