Curriculum
32 topics across 8 tracks — from probability foundations to high-dimensional statistics.
Every topic connects backward to formalCalculus prerequisites and forward to formalML topics it enables.
Prerequisite Graph
The full dependency graph — arrows show prerequisites. Filled nodes are published topics.
Foundations of Probability
Kolmogorov axioms, conditional probability, random variables, expectation
Conditional Probability & Independence
Bayes' theorem, law of total probability, conditional independence
Random Variables & Distributions
Measurable functions, PMFs and PDFs, CDFs
Expectation, Variance & Moments
Integration against a measure, moment-generating functions, characteristic functions
Core Distributions & Families
Discrete and continuous distributions, exponential families, multivariate distributions
Discrete Distributions
Bernoulli, Binomial, Poisson, Geometric, Negative Binomial
Continuous Distributions
Normal, Exponential, Gamma, Beta, Uniform
Exponential Families
Sufficient statistics, natural parameters, log-partition function
Multivariate Distributions
Joint, marginal, conditional densities, the multivariate normal
Convergence & Limit Theorems
Modes of convergence, law of large numbers, central limit theorem, tail bounds
Modes of Convergence
Almost sure, in probability, in distribution, in Lp
Law of Large Numbers
Weak and strong LLN, Kolmogorov's theorem
Central Limit Theorem
Lindeberg-Lévy, Berry-Esseen bound
Large Deviations & Tail Bounds
Markov, Chebyshev, Chernoff, Hoeffding, sub-Gaussian theory
Statistical Estimation
Bias-variance, maximum likelihood, method of moments, sufficiency
Point Estimation & Bias-Variance
Estimators as random variables, bias, variance, MSE decomposition
Maximum Likelihood Estimation
Likelihood function, score, Fisher information, asymptotic normality
Method of Moments & M-Estimation
Moment equations, generalized method of moments, Z-estimators
Sufficient Statistics & Rao-Blackwell
Information compression, UMVUE, completeness
Hypothesis Testing & Confidence
Neyman-Pearson paradigm, likelihood ratio tests, confidence intervals, multiple testing
Hypothesis Testing Framework
Null and alternative, Type I/II errors, power, p-values
Likelihood Ratio Tests & Neyman-Pearson
Most powerful tests, the likelihood ratio principle, Wilks' theorem
Confidence Intervals & Duality
Pivotal quantities, inversion of tests, coverage probability
Multiple Testing & False Discovery
Bonferroni, Holm, Benjamini-Hochberg FDR
Regression & Linear Models
Least squares, generalized linear models, regularization, model selection
Simple & Multiple Linear Regression
Least squares as projection, Gauss-Markov theorem, residual analysis
Generalized Linear Models
Link functions, exponential family connection, deviance
Regularization & Penalized Estimation
Ridge, lasso, elastic net — bias-variance as explicit penalization
Model Selection & Information Criteria
AIC, BIC, cross-validation theory, Mallows' Cp
Bayesian Statistics
Prior selection, MCMC computation, model comparison, hierarchical models
Bayesian Foundations & Prior Selection
Prior, likelihood, posterior, conjugacy, Jeffreys priors
Bayesian Computation
MCMC, Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo
Bayesian Model Comparison
Bayes factors, marginal likelihood, posterior predictive checks
Hierarchical & Empirical Bayes
Multilevel models, shrinkage estimators, James-Stein
High-Dimensional & Nonparametric
Order statistics, kernel density estimation, bootstrap, empirical processes
Order Statistics & Quantiles
Distribution-free inference, sample quantile asymptotics
Kernel Density Estimation
Bandwidth selection, bias-variance for density estimation
The Bootstrap
Efron's bootstrap, bootstrap confidence intervals, bootstrap hypothesis tests
Empirical Processes & Uniform Convergence
Glivenko-Cantelli, Donsker's theorem, VC dimension