Math for Causal Inference
PHASE 1 · DAYS 1–15

Probability & Statistical Inference

Axioms, conditional probability, expectations, distributions, the CEF, LLN/CLT, estimators, intervals, tests, MLE, the delta method, Bayes and missing data.

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  1. 1
    Probability Axioms & Set Theory

    Sample spaces, events and Kolmogorov's three axioms, and the rules that follow from them.

    Inclusion–Exclusion & Fréchet Bounds
  2. 2
    Conditional Probability, Independence & Bayes

    Updating probabilities when you learn something, and the four rules that do it.

    Bayes' Theorem Calculator
  3. 3
    Random Variables, Expectation & Variance

    Turning outcomes into numbers, and the two summaries every estimator is built from.

    Linearity of Expectation & Variance Scaling
  4. 4
    Common Distributions & the Data-Generating Process

    The handful of distributions that model most economic data, and how to write a DGP with potential outcomes.

    Distribution Explorer & DGP Sampler
  5. 5
    Joint Distributions, Covariance & Correlation

    How two variables move together, and why that is the raw material of regression and omitted-variable bias.

    Bivariate Normal in 3D
  6. 6
    Conditional Expectation, the LIE & Total Variance

    The conditional expectation function, the law of iterated expectations and the adjustment formula.

    Conditional Expectation & Total Variance
  7. 7
    The Law of Large Numbers & Central Limit Theorem

    Why sample means settle down, and why their errors are approximately normal.

    Central Limit Theorem Simulator
  8. 8
    Estimators: Bias, Variance, MSE & Consistency

    How to judge an estimator, and why "unbiased" is not the same as "good".

    Estimator Race: Bias, Variance & MSE
  9. 9
    Confidence Intervals & the Bootstrap

    Turning a standard error into an interval with a known coverage rate, analytically or by resampling.

    Confidence-Interval Coverage
  10. 10
    Hypothesis Testing, p-Values & Power

    Deciding whether an effect is distinguishable from zero, and designing studies that can tell.

    Statistical Power Visualiser
  11. 11
    Maximum Likelihood Estimation

    Choosing the parameter that makes the observed data most probable, and measuring its precision from the curvature.

    Log-Likelihood Explorer
  12. 12
    The Delta Method

    Standard errors for functions of estimates (logs, ratios, odds, percentage effects) via a first-order Taylor expansion.

    Delta Method vs Simulation
  13. 13
    Bayesian Inference

    Priors, likelihoods and posteriors; conjugate updating, shrinkage and credible intervals.

    Beta–Binomial Updating
  14. 14
    Missing Data & Inverse Probability Weighting

    MCAR, MAR and MNAR; when complete-case analysis is biased; and how weighting repairs MAR.

    Missing-Data Mechanisms
  15. 15
    Phase 1 Capstone: Simpson's Paradox, Conditional Independence & Selection Bias

    Pulling Phase 1 together: when do comparisons in data answer causal questions?

    Simpson's Paradox Lab