Causal Inference for Analysts

A practical guide to causal reasoning using Judea Pearl's framework of structural causal models, do-calculus, and counterfactual analysis—with applications to Marketing Mix Modeling.

Why Causal Inference?

Marketing Mix Modeling asks fundamentally causal questions: What is the effect of TV advertising on sales? If we had spent more on digital, how much more would we have sold? These questions cannot be answered by correlation alone.

The goal of causal inference is to answer interventional and counterfactual questions from observational data—when experiments are impossible or impractical.

The Fundamental Problem of Causal Inference

Consider a single week where we spent 100K USD on TV. We observed sales of 1M USD. What would sales have been if we had spent 0 USD on TV that week? We can never observe this directly— we can't rewind time and run the alternative scenario.

Where \(Y^{a=1}\) is the potential outcome under treatment and \(Y^{a=0}\) is the potential outcome under control. We observe one; the other is the counterfactual.

Structural Causal Models

A Structural Causal Model (SCM) is a mathematical object that encodes causal relationships. It consists of three components:

Example: Simple Marketing SCM

The structural equations are asymmetric: \(\text{AdSpend} = f(\text{Budget})\) means Budget causes AdSpend, not the reverse. This asymmetry encodes the direction of causation.

Directed Acyclic Graphs (DAGs)

Every SCM implies a Directed Acyclic Graph (DAG) where nodes are variables and edges point from causes to effects. DAGs provide a visual language for causal reasoning.

d-Separation

d-separation is a graphical criterion for reading conditional independence from a DAG. Two variables are d-separated (conditionally independent) given a set Z if every path between them is "blocked."

The Three Causal Building Blocks

Pearl's Ladder of Causation

Pearl's Ladder of Causation (also called the Causal Hierarchy) describes three levels of causal reasoning, each requiring different information and enabling different queries.

Level 1: Association

Association queries ask about statistical dependencies: How are X and Y related in the data?

Standard machine learning and predictive modeling operate at Level 1. Given historical data, they learn \(P(Y|X)\)—the distribution of Y given we observe X. This is sufficient for prediction but not for intervention.

Level 2: Intervention

Intervention queries ask what happens when we act: If I set X to x, what happens to Y?

The \(do(\cdot)\) operator represents an intervention—physically setting a variable to a value, rather than passively observing it. This is fundamentally different from conditioning.

Level 3: Counterfactuals

Counterfactual queries ask about alternative histories: Given what actually happened, what would have happened under different circumstances?

This asks: "For a unit that actually received X=x and had outcome Y=y, what would Y have been if X had been x' instead?" This is the language of individual causal effects.

The do-Operator

The \(do(\cdot)\) operator formalizes intervention. When we write \(do(X=x)\), we mean:

1. Delete all arrows pointing into X in the causal graph
2. Set X to the value x
3. Compute the resulting distribution over other variables

This "graph surgery" reflects what happens in an experiment: we override the natural causes of X and force it to take a specific value.

The Three Rules of do-Calculus

Pearl's do-calculus provides three rules for manipulating expressions containing \(do(\cdot)\). Together, they are complete: any identifiable causal effect can be computed using these rules.

Identification

A causal effect is identifiable if it can be computed from observational data plus the causal graph structure, without knowing the functional forms \(f_i\).

The Backdoor Criterion

The backdoor criterion provides a simple sufficient condition for identification.

If Z satisfies the backdoor criterion, the causal effect is identified by the backdoor adjustment formula:

The Frontdoor Criterion

When the backdoor criterion fails (no valid adjustment set exists), the frontdoor criterion may still identify the effect through a mediator.

Instrumental Variables

An instrumental variable Z affects X but has no direct effect on Y (except through X). It provides identification when direct backdoor adjustment is impossible.

For linear models, the IV estimate is:

Computing Counterfactuals

Counterfactual reasoning requires three steps:

Example: Counterfactual Sales

Consider a simple linear SCM:

Observation: We observed X=100 (spent 100K USD) and Y=500 (500K USD sales).

Question: What would Y have been if X had been 0?

Step 1 (Abduction): From Y = βX + U_Y and the observation, infer U_Y = 500 - 100β.

Step 2 (Action): Set X = 0 in the modified model.

Step 3 (Prediction): Y_{X=0} = β(0) + U_Y = 500 - 100β.

If β = 3 (each 1K USD in ad spend generates 3K USD in sales), then Y_{X=0} = 500 - 300 = 200K USD.

Common Pitfalls

Confounding

Confounding occurs when a common cause of treatment and outcome creates a spurious association.

Solution: Control for confounders via backdoor adjustment, or use experimental designs.

Collider Bias

Collider bias (selection bias, Berkson's paradox) occurs when you condition on a common effect of two variables, creating a spurious association between them.

Overcontrol Bias

Overcontrol bias occurs when you adjust for a mediator on the causal pathway, blocking part of the effect you're trying to measure.

Rule: Only control for confounders, never mediators (unless you specifically want the direct effect).

MMM as a DAG

A Marketing Mix Model encodes specific causal assumptions. Here's the DAG implied by a standard MMM:

Media Effect Identification

For media effects to be identified in an MMM, we need to satisfy the backdoor criterion by controlling for all common causes of media spend and sales.

Counterfactual Contribution Analysis

MMM contribution analysis is fundamentally counterfactual: "How much of observed sales is attributable to each channel?" This requires computing:

This is the difference between actual sales and the counterfactual sales if channel c had been set to zero. The MMM Framework implements this via:

# Counterfactual contribution in the framework
contributions = model.compute_contributions(
    method="counterfactual",  # vs "marginal" decomposition
    baseline="zero"           # counterfactual: set channel to zero
)

# For each posterior draw, compute Y - Y_{do(X_c=0)}
# Returns full posterior distribution of contributions

Specification Shopping and Causal Inference

Specification shopping—trying multiple model specifications and selecting the one that gives "reasonable" results—is incompatible with causal inference.

The Correct Approach

1. Specify the DAG first — Based on domain knowledge, not data
2. Identify adjustment sets — Use backdoor/frontdoor criteria
3. Specify priors — Encode beliefs before seeing results
4. Fit once and report — Accept wide uncertainty if that's what the data support
5. Iterate scientifically — Model changes must be justified by theory, not by making results look better

Key Takeaways

References

• Pearl, J. (2009). Causality: Models, Reasoning, and Inference (2nd ed.). Cambridge University Press.
• Pearl, J., Glymour, M., & Jewell, N. P. (2016). Causal Inference in Statistics: A Primer. Wiley.
• Pearl, J., & Mackenzie, D. (2018). The Book of Why: The New Science of Cause and Effect. Basic Books.
• Cunningham, S. (2021). Causal Inference: The Mixtape. Yale University Press. [Free online]
• Hernán, M. A., & Robins, J. M. (2020). Causal Inference: What If. Chapman & Hall/CRC. [Free online]
• Peters, J., Janzing, D., & Schölkopf, B. (2017). Elements of Causal Inference. MIT Press.