Counterfactual Analysis with SCMS

1. The Mechanics of Counterfactual Computation

As we previewed, computing a unit-level counterfactual Y(t_cf) given a factual observation Y(t_fact) = y_fact requires a fully specified SCM. Let's revisit the three-step process, which is the engine of counterfactual inference.

1. Abduction: Use the factual evidence (t_fact, y_fact) to update your knowledge about the exogenous variables U. These U variables represent the stable, unobserved characteristics of the system—the triad's unique "relational physics." In an invertible model, you can solve for U exactly. In a non-invertible model, you derive a posterior distribution, P(U | evidence).
2. Action: Modify the SCM to represent the counterfactual world. This is achieved by applying the do-operator, replacing the structural equation for the treatment variable with its counterfactual value, T := t_cf.
3. Prediction: Use the inferred knowledge of U from Step 1 and the modified model from Step 2 to compute the counterfactual outcome.

The following brief article section provides a crisp mathematical summary of this process with a simple example.

2. A Deterministic Example: Linear SCM for the Vee Triad

Let's apply this process to a simplified model of your vee triad. Assume the following causal structure, where the pivot A's satisfaction directly influences the satisfaction of partners B and C, and the living arrangement T affects everyone.

• Units: Pivot A, Partner B, Partner C.
• Treatment T: A binary variable where T=1 (all live together) and T=0 (all live separately).
• Exogenous Us: U_A, U_B, U_C are latent, normally distributed variables representing each person's baseline satisfaction or disposition. U_A, U_B, U_C \sim \mathcal{N}(0, 1).
• Outcomes Ys: Y_A, Y_B, Y_C are the continuous satisfaction scores for each member.

We can specify a linear SCM as follows:

Here, the \alpha coefficients represent the direct effect of the living arrangement, and the \gamma coefficients model the interference—how A's satisfaction spills over to B and C. Let's assume we have estimated these parameters from prior data (a topic for a later module) and found: \alpha_A=2, \alpha_B=1, \alpha_C=1, \gamma_{AB}=0.5, \gamma_{AC}=0.4.

Now, let's compute a counterfactual.

Factual Scenario: The triad tries living separately (T=0), and you observe their satisfaction scores: Y_A = -0.5, Y_B = 1.0, Y_C = -0.8.

Counterfactual Question: What would their satisfaction scores have been if they had all lived together (T=1)?

Let's execute the three steps:

1. Abduction: We use the factual observations to solve for the specific values of U_A, U_B, U_C for this triad.

This abduction step has given us a window into the soul of this triad. We've inferred their latent characteristics (U_A=-0.5, U_B=1.25, U_C=-0.6). These values are fixed for this specific triad.

2. Action: We modify the SCM to represent the world where they live together by setting do(T=1). The equations become:

3. Prediction: We now predict the counterfactual outcomes by plugging in the inferred U values.

Our computation concludes that, for this specific triad with their unique latent traits, had they chosen to live together, their satisfaction scores would have been (Y_A=1.5, Y_B=3.0, Y_C=1.0).

3. Probabilistic Counterfactuals: The Non-Invertible Case

The linear model was convenient because the relationship between outcomes and latent variables was invertible, allowing us to pinpoint the exact values of U_A, U_B, U_C. In reality, this is rarely the case. For instance, if your satisfaction outcome Y is measured on a discrete 1-7 Likert scale, the function Y = f(U) is no longer invertible. Multiple values of a continuous latent variable U could map to the same discrete outcome Y=5.

When we cannot uniquely identify U, we cannot compute a single, deterministic counterfactual. Instead, we compute a probabilistic counterfactual: a probability distribution over the possible counterfactual outcomes. Your background in Bayesian methods is directly applicable here. The abduction step changes from solving an equation to performing Bayesian updating.

1. Abduction (Probabilistic): Start with a prior distribution for U, P(U). Use the factual evidence E (e.g., T=0, Y_A=5) to compute the posterior distribution P(U|E) \propto P(E|U)P(U).
2. Action: Modify the model to do(T=1).
3. Prediction (Probabilistic): Propagate the posterior P(U|E) through the modified SCM to get a distribution for the counterfactual outcome P(Y(T=1) | E).

The video from our previous lesson explains this transition from a deterministic to a probabilistic result very clearly. A quick review will solidify the concept.

4. A Framework for Categorical Outcomes: Gumbel-Max SCMs

The probabilistic approach is powerful, but it requires us to choose a specific parametric form for our SCM. For categorical outcomes like your living arrangements (t_1, t_2, t_3) or Likert-scale satisfaction scores, what is a principled choice?

One elegant and powerful framework is the Gumbel-Max Structural Causal Model. This approach, common in discrete choice modeling, assumes that an agent selects the category j that maximizes a utility function log(p_j) + g_j, where p_j is the baseline probability of that category and g_j is an unobserved shock drawn from a Gumbel distribution.

In your context, you can model a person's satisfaction level as a "choice" from k possible levels. The probability of landing on level j depends on the treatment T and their latent Gumbel noise g_j.

• SCM: Y = argmax_j {log(P(Y=j|T)) + g_j}
• Abduction: When we observe Y=i under treatment T, we don't know the exact g values. But we know that log(P(Y=i|T)) + g_i was the highest utility. This constrains the possible values of g, yielding a posterior P(g | Y=i, T).
• Action: We introduce a counterfactual treatment T', which changes the probabilities to P(Y=j|T').
• Prediction: We take samples from the posterior P(g | Y=i, T), add them to the new log-probabilities log(P(Y=j|T')), and find the new argmax. The distribution of these new argmaxes is our probabilistic counterfactual outcome.

This method is not just a computational trick; it has desirable theoretical properties. The following sections from Michael Oberst's thesis provide the formal details and a practical application. Given your background in stochastic processes and simulation, the idea of generating counterfactual trajectories by sampling from a posterior will feel very natural.

The Gumbel-Max SCM provides a ready-made, theoretically sound engine for computing the counterfactuals central to your project. By simulating trajectories from the posterior, you can generate distributions of what might have happened for each member of the triad under each alternative living arrangement.

Conclusion

Today, we have transitioned from formulating counterfactual questions to computing their answers. You now have a complete toolkit for this task, grounded in the three-step algorithm of abduction, action, and prediction.

Key Takeaways:

• Computing unit-level counterfactuals requires a fully specified parametric SCM.
• In invertible (e.g., linear) models, you can perform abduction by solving for the exact values of the latent U variables, yielding a single, deterministic counterfactual outcome.
• In non-invertible (e.g., discrete outcome) models, abduction involves computing a posterior distribution over U, resulting in a probabilistic counterfactual outcome.
• The Gumbel-Max SCM offers a principled and practical framework for computing probabilistic counterfactuals for categorical variables, which is highly relevant for modeling choices and satisfaction levels in your triadic experiment.

In our next lesson, we will put these computed counterfactuals to use. You will learn how to leverage them to decompose the total causal effect of a living arrangement on an individual's satisfaction into its direct and indirect (spillover) components, allowing you to untangle the complex web of influence within the triad.