Causal Models for Counterfactuals

1. The Realm of Imagination: What are Counterfactuals?

Counterfactuals are statements about worlds that could have been. They are subjunctive conditionals that ask what would have happened had circumstances been different.

• "If I had not taken the aspirin, would my headache still be gone?"
• "Was it the drug that caused this patient's recovery, or would they have recovered anyway?"
• In the context of your research, "If the couple had not entered therapy, would they have separated?"

These questions cannot be answered by observation or even by intervention alone. Answering them requires us to consider a world that conflicts with the one we observed (the patient did take the aspirin) and to somehow transport information from the factual world to this hypothetical one.

2. From Causal Graphs to Causal Models

In previous lessons, we used causal DAGs to represent our assumptions. However, to formally answer counterfactual questions, a DAG combined with a probability distribution is not enough. We need to specify the mechanisms of the system. This is the role of a Structural Causal Model (SCM).

Recall from our previous discussion that an SCM augments a DAG with a set of equations, specifying precisely how each variable is determined by its causes.

The crucial philosophical point is this: two different SCMs can imply the exact same observational data ($P(V)$) and the exact same interventional effects ($P(Y|\text{do}(X))$), yet yield completely different answers to counterfactual questions. This means that counterfactuals allow us to "see" a deeper layer of the causal reality.

This reveals a profound point about causal knowledge. The functional form of the SCM, particularly the nature of the unobserved background variables $U$, represents the underlying causal mechanisms. Counterfactuals are queries about these mechanisms, which are often invisible to lower-level analysis.

3. The Algorithm for Imagination: Abduction, Action, Prediction

So, how does an SCM provide a formal language for counterfactuals? By enabling a three-step reasoning process:

1. Abduction: Use the factual evidence from the real world to update your belief about the unobserved background variables ($U$). This is a classic Bayesian step: given the observed evidence, what is the posterior probability distribution over the latent "type" of the individual unit?
2. Action: Perform a "surgical" intervention on the model to create the hypothetical world. This is identical to the do-operator: replace the structural equation for the antecedent variable.
3. Prediction: Use the modified model and the updated belief about $U$ from Step 1 to calculate the probability of the consequent in the hypothetical world.

Now, let's see this algorithm in action.

As you can see, the SCM provides the formal structure needed to execute this three-step process. The exogenous variables $U$ represent the unit-specific characteristics that determine how an individual responds to a cause. Abduction is the process of inferring those characteristics from observed behavior.

4. Why Counterfactuals Matter: Digging Deeper

The ability to compute counterfactuals is not just an academic exercise. It provides a more refined understanding of causality that can be essential for explanation, credit assignment, and fairness.

Consider the example from the UCLA video where two models had the same average causal effect but different underlying mechanisms. Interventional queries ($P(Y|\text{do}(X))$) could not tell them apart. But counterfactual queries could.

These examples demonstrate that counterfactuals are the key to moving from "What is the average effect of X on Y in the population?" to "For this specific individual who experienced outcome Y after being exposed to X, was it X that caused Y?" This is the essence of explanation and attribution.

The twin network model (mentioned in the UCLA video around the 31-minute mark) is a useful mental model for this. Imagine the factual world and the counterfactual world as two parallel universes. They are identical except for the intervention. The background variables $U$ are shared between them, acting as the bridge that carries information from the facts we observed to the hypothetical world we are reasoning about.

Conclusion

Today we reached the summit of the Ladder of Causation. We have seen that SCMs provide the necessary formal language to answer counterfactual questions, something that associational and interventional models alone cannot do.

Key Takeaways:

• Counterfactuals (L3) are "what if" questions about past events, involving reasoning about scenarios that conflict with reality (e.g., "what if the patient hadn't taken the drug, given that they did?").
• SCMs are Essential: Answering counterfactuals requires knowledge of the underlying causal mechanisms, which are formalized by the equations and background variables ($U$) in a Structural Causal Model. Lower-level information (from L1 and L2) underdetermines counterfactuals.
• The Three-Step Algorithm: Counterfactuals are computed via Abduction (updating beliefs about $U$ given evidence), Action (modifying the SCM), and Prediction (calculating the outcome in the modified model).
• A Deeper Understanding: Counterfactual reasoning allows for a more refined, unit-level analysis of causation, enabling explanations and attributions that are impossible with interventionist or associational queries alone.

This lesson completes our journey up the causal hierarchy. We've seen how a fully specified SCM allows us to answer questions at all three levels. In our next lesson, we will address a crucial question: What if we don't know the SCM? We will explore how Bayesian methods can be used to learn about the structure of the causal model itself when we are uncertain.