Statistical Inference

Notes & simulations

Interactive simulations for statistics, econometrics, and causal inference. The idea is to see how things like sampling distributions, p-values, and OLS actually work, not just read about them.

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Probability & Uncertainty

• Foundations — Distributions, Sampling & Confidence Intervals
• Variance, SD & Standard Error — Population vs Sample, SD vs SE & Why n − 1
• The Sampling Distribution — Data vs Sampling Distribution & When to Assume Normality

The Central Limit Theorem

• CLT Simulator — Interactive CLT Simulator
• LLN vs CLT — Why Averages Stabilize Before They Become Normal

Inference

• p-values & Confidence Intervals — What They Actually Mean
• Test Statistics — Z, t, F, Chi-Squared, Wald, LR & Score
• The Bootstrap — Resampling-Based Inference
• Power, Alpha, Beta & MDE — Hypothesis Testing & Experiment Design
• Monte Carlo Experiments — How We Understand Estimators
• Multiple Testing — False Discoveries & the Replication Crisis

Regression

• Regression & the CEF — OLS and the Conditional Expectation Function
• Residuals & Controls — Diagnostics, Partialling Out & Omitted Variable Bias
• Frisch-Waugh-Lovell — Partialling Out & OVB
• Omitted Variable Bias — The OVB formula, sign-of-bias table & when controls eliminate confounding

Algebra of Regression

• The Algebra Behind OLS — Matrix notation, the VCV matrix, and where standard errors come from

Estimation

• Method of Moments — Match sample moments to population moments
• Maximum Likelihood — Find the parameters that make the data most probable
• Limited Dependent Variables — Logit, probit, marginal effects & the Tobit model
• Generalized Method of Moments — When you have more moment conditions than parameters
• Bayesian Estimation — MAP, posterior mean, and the link to regularization

Standard Errors & Diagnostics

• Heteroskedasticity — Constant vs Non-Constant Variance & Robust SEs
• Clustered SEs — When Observations Aren’t Independent
• The Delta Method — Standard errors for nonlinear transformations of estimates
• Measurement Error — Why Noisy Regressors Bias You Toward Zero
• Bias-Variance Tradeoff — Underfitting, Overfitting & MSE Decomposition
• Model Selection — AIC, BIC, cross-validation & choosing the right complexity
• When Inference Breaks Down — Why Variation Is the Fuel of Inference

Causal Thinking

• From Correlation to Causation — Why Correlation Isn’t Enough & When It Becomes Causal

Bayesian Thinking

• Bayesian Updating — One Gentle Introduction

Statistical Foundations of AI

• Training as Maximum Likelihood — Cross-entropy, SGD, and what training optimizes
• Regularization as Bayesian Inference — Weight decay, dropout, and the RLHF penalty as priors
• Prediction vs Causation in Foundation Models — \(P(Y \mid X)\) vs \(P(Y \mid do(X))\) and identification vs training
• Experimental Design for AI Systems — A/B testing, power, and multiple testing for model evaluation
• Calibration & Uncertainty Quantification — When to trust model confidence