Causal Inference Basics

Most questions a product or business stakeholder actually asks are causal: did the marketing campaign bring in customers, did the new feature cause retention to rise, would this user have churned without the discount. Most data available to answer them is observational: the campaign targeted people who were already likely to buy, the feature was adopted by the most engaged users, the discount went to people who complained. Naive comparisons of treated and untreated groups answer a different question — "who got treated?" — and answer it confidently and wrongly.

Randomised experiments (the A/B Testing & Online Experimentation subject) are the clean solution, but a large share of real decisions cannot be randomised: company-wide launches, pricing visible to everyone, marketing channels with no clean control, regulatory changes, historical decisions someone now wants evaluated. Causal inference is the discipline of getting a defensible answer anyway — and, just as importantly, of knowing when no defensible answer exists.

This subject gives you the vocabulary (potential outcomes, ATE, confounders, colliders, back-door), the standard observational designs each with a small numeric example, the assumptions each design rests on, the checks you can run, and the ways these methods fail in practice. It is the difference between a data scientist who says "adopters retain 20% better" and one who says "after adjusting for pre-adoption engagement, and under the assumption that no unmeasured factor drives both adoption and retention, the feature raised retention by about 4 points; here is how sensitive that is to a hidden confounder."

Potential Outcomes and the Fundamental Problem

The Rubin causal model defines causation per unit. Each unit i has two potential outcomes:

• Y_i(1) — the outcome if treated
• Y_i(0) — the outcome if not treated

The individual causal effect is \tau_i = Y_i(1) - Y_i(0). The fundamental problem of causal inference: you observe exactly one of the two, never both. The other is the counterfactual.

unit  treated?  Y(1)   Y(0)   observed Y   effect τ
 A       1       10     ?         10          ?
 B       1        7     ?          7          ?
 C       0        ?     6          6          ?
 D       0        ?     3          3          ?

Because individual effects are unidentifiable, we target averages:

• ATE — average treatment effect over the whole population: E[Y(1) - Y(0)]
• ATT — average effect on the treated: E[Y(1) - Y(0) \mid D = 1]. Often the policy-relevant quantity ("what did the campaign do for the people who received it?").
• CATE — conditional on covariates X: E[Y(1) - Y(0) \mid X = x]. The target of uplift modelling.

The naive comparison E[Y \mid D=1] - E[Y \mid D=0] decomposes as

The second term is the difference in untreated outcomes between the people who got treated and the people who did not. If treated users would have retained better anyway, selection bias is positive and the naive estimate over-states the effect. Randomisation sets this term to zero in expectation; every observational method is an attempt to make it zero by assumption.

The identifying assumptions all methods draw on:

• Ignorability / unconfoundedness: \{Y(0), Y(1)\} \perp D \mid X — given the covariates you adjust for, treatment is as good as random.
• Overlap / positivity: 0 < P(D=1 \mid X) < 1 — every kind of unit has some chance of both conditions. Without overlap you are extrapolating, not comparing.
• SUTVA: no interference between units, one version of treatment.