Introduction to Causal Inference#

Why causality?#

We constantly reason about “what caused what.” If an ad runs and sales rise, was it the ad—or something else? Causal inference turns this intuition into a careful, testable framework so we can answer policy and product questions with data.

The core idea#

For any unit (a person, store, app session, etc.) we imagine two potential outcomes:

  • \(Y(1)\): what would happen with the treatment

  • \(Y(0)\): what would happen without the treatment

We only ever observe one of them. This “missing counterfactual” is the fundamental challenge of causal inference. Everything we do is about approximating the missing outcome in a principled way.

Notation#

  • Units: \(i=1,\dots,n\)

  • Treatment: \(T_i\in\{0,1\}\) (can be multi-valued or continuous)

  • Outcome: \(Y_i\) (what we observe)

  • Covariates / confounders: \(X_i\) (pre-treatment features that affect both treatment and outcome)

Example dataset:

Unit

Treatment \(T\)

Outcome \(Y\)

Confounder_1

Confounder_2

1

1

100

1

13

2

0

90

0

14

3

1

110

1

51

4

1

97

1

63

5

0

80

0

34

6

0

85

0

53

What we aim to estimate (estimands)#

ITE / CATE: individual or subgroup effect#

  • Individual Treatment Effect (ITE): \(\text{ITE}_i = Y_i(1)-Y_i(0)\) (unobservable for a single unit)

  • Conditional Average Treatment Effect (CATE): \(\tau(x)=\mathbb{E}[Y(1)-Y(0)\mid X=x]\) Use when you need personalization, uplift targeting, or to study heterogeneity.

ATT: effect on the treated#

\(\text{ATT}=\mathbb{E}[Y(1)-Y(0)\mid T=1]\) Use to answer “Did it work for those who actually received it?”—e.g., after a selective rollout.

ATE: overall average effect#

\(\text{ATE}=\mathbb{E}[Y(1)-Y(0)]\) Use for headline impact or policy choices that affect the whole eligible population.

When are these estimands credible?#

In observational data, we typically rely on two key identification conditions:

  1. Unconfoundedness (selection on observables): given \(X\), treatment is as good as random. Formally, \((Y(0),Y(1)) \perp T \mid X\).

  2. Overlap (positivity): each unit had a nonzero chance to receive either treatment: \(0<\Pr(T=1\mid X)<1\).

These assumptions won’t be true by magic; we design models and diagnostics to make them as plausible as possible.

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