1. Motivation

• Randomized Controlled Trial (RCT): the gold standard for causal effect estimation (e.g. BNT162b2 vaccine vs. placebo)
• Observational Studies: require adjustment for confounders (age, sex, residence, chronic conditions, PCR tests) to estimate treatment effects (e.g. booster effectiveness)
• Spurious Correlations: “correlation ≠ causation” (e.g. Tyler Vigen examples)
• Prediction ≠ Causation: supervised models can mislead (e.g. asthma history and pneumonia mortality)
• Confounding Examples:
  • Smoking and lung cancer (ethical RCT impossible; observational comparisons confounded)
  • Simpson’s paradox in kidney stone treatment (aggregate vs. stratified success rates)

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2. Causality

• Practical Definition: \(T\) causes \(Y\) if intervening on \(T\) changes \(Y\), holding other factors constant
• Do-Operator: \(P(Y \mid \text{do}(T))\) denotes the distribution of \(Y\) under an intervention on \(T\)
• Causal Graphs (DAGs): nodes = variables; edges = direct causal effects
  • Example: Aspirin → Stroke, with confounders (e.g. coronary heart disease)

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3. Conditioning

• Back-Door Adjustment:
  \(P(Y \mid \text{do}(T)) = \sum_x P(Y \mid T, X = x)\,P(X = x)\)
• Resolving Simpson’s Paradox: stratify by stone size to recover the true causal effect in kidney stone treatment
• Example: impact of stationary biking on cholesterol is confounded by age; conditioning on age blocks the back-door path

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4. Confounding, Mediation & Colliding

• Confounder: common cause of treatment and outcome; conditioning blocks bias
• Mediator: lies on the causal path; conditioning may remove part of the effect
• Collider: common effect of treatment and outcome; conditioning opens a spurious association (Berkson’s paradox)

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5. Potential Outcomes

• Individual Potential Outcomes: \(Y_i(A)\), \(Y_i(B)\) under treatments \(A,B\)
• Individual Treatment Effect (ITE): \(Y_i(A) - Y_i(B)\)
• Average Treatment Effect (ATE): \(\mathbb{E}[Y_i(A) - Y_i(B)]\)
• Fundamental Problem: only one potential outcome is observed per unit; counterfactual must be estimated \(\mathbb{E}[Y_i(A)-Y_i(B)] \ne \mathbb{E}[Y_i \mid T=A]- \mathbb{E}[Y_i \mid T=B]\) —

6. Counterfactual Inference Approaches

1. Covariate Adjustment
   • Regression (model \(Y\sim T + X\); sensitive to model misspecification, multicollinearity)
   • Matching (pair treated/untreated on \(X\))
     • \[\text{match}(i)=\min_j \text{Distance}(X_i, X_j)\]
     • only works well when matched pairs are very similar
   • Stratification (group by \(X\)-levels)
2. Propensity Score \(e(X)=P(T=1\mid X)\)
   • Estimated via logistic regression
   • Use for matching, stratification, or Inverse Probability of Treatment Weighting (IPTW)
   • Balances covariates across treatment groups; high‐variance if \(e(X)\) near 0 or 1
   • Always one dimensional even if X is high-dimensional
   • If conditioning on X suffices to control for all confounding, so does conditioning on estimated PS

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7. Summary

• Two ML‐based causal inference paradigms:
  1. Covariate‐based outcome modeling
  2. Propensity‐based reweighting/stratification
• Valid inference relies on assumptions: unconfoundedness, positivity, consistency, no interference

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