Curriculum

32 topics across 9 tracks (3 planned) — 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.

ProbabilityDistributionsConvergenceEstimationTestingRegressionBayesianNonparametricDrag nodes · Scroll to zoom

Foundations of Probability

Kolmogorov axioms, conditional probability, random variables, expectation

foundational

Sample Spaces, Events & Axioms

Kolmogorov axioms, sigma-algebras for the working statistician, combinatorial probability

foundational

Conditional Probability & Independence

Bayes' theorem, law of total probability, conditional independence

foundational

Random Variables & Distribution Functions

Measurable functions, PMFs and PDFs, CDFs

foundational

Expectation, Variance & Moments

Integration against a measure, moment-generating functions, characteristic functions

Core Distributions & Families

Discrete and continuous distributions, exponential families, multivariate distributions

foundational

Discrete Distributions

Bernoulli, Binomial, Poisson, Geometric, Negative Binomial

foundational

Continuous Distributions

Normal, Exponential, Gamma, Beta, Uniform

intermediate

Exponential Families

Sufficient statistics, natural parameters, log-partition function

intermediate

Multivariate Distributions

Joint, marginal, conditional densities, the multivariate normal

Convergence & Limit Theorems

Modes of convergence, law of large numbers, central limit theorem, tail bounds

intermediate

Modes of Convergence

Almost sure, in probability, in distribution, in Lp

intermediate

Law of Large Numbers

Weak and strong LLN, Kolmogorov's theorem

intermediate

Central Limit Theorem

Lindeberg-Lévy, Berry-Esseen bound

advanced

Large Deviations & Tail Bounds

Markov, Chebyshev, Chernoff, Hoeffding, sub-Gaussian theory

Statistical Estimation

Bias-variance, maximum likelihood, method of moments, sufficiency

foundational

Point Estimation & Bias-Variance

Estimators as random variables, bias, variance, MSE decomposition

intermediate

Maximum Likelihood Estimation

Likelihood function, score, Fisher information, asymptotic normality

intermediate

Method of Moments & M-Estimation

Moment equations, generalized method of moments, Z-estimators

intermediate

Sufficient Statistics & Rao-Blackwell

Information compression, UMVUE, completeness

Hypothesis Testing & Confidence

Neyman-Pearson paradigm, likelihood ratio tests, confidence intervals, multiple testing

foundational

Hypothesis Testing Framework

Null and alternative, Type I/II errors, power, p-values

intermediate

Likelihood-Ratio Tests & Neyman-Pearson

intermediate

Confidence Intervals & Duality

intermediate

Multiple Testing & False Discovery

Regression & Linear Models

Least squares, generalized linear models, regularization, model selection

foundational

Simple & Multiple Linear Regression

Least squares as projection, Gauss-Markov theorem, residual analysis

intermediate

Generalized Linear Models

Link functions, exponential family connection, deviance

intermediate

Regularization & Penalized Estimation

Ridge, lasso, elastic net — bias-variance as explicit penalization

intermediate

Model Selection & Information Criteria

AIC, BIC, cross-validation theory, Mallows' Cp

Bayesian Statistics

Prior selection, MCMC computation, model comparison, hierarchical models

intermediate

Bayesian Foundations & Prior Selection

Prior, likelihood, posterior, conjugacy, Jeffreys priors

intermediate

Bayesian Computation & MCMC

MCMC, Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo

advanced

Bayesian Model Comparison & BMA

Bayes factors, marginal likelihood, posterior predictive checks

advanced

Hierarchical & Empirical Bayes

Multilevel models, shrinkage estimators, James-Stein

High-Dimensional & Nonparametric

Order statistics, kernel density estimation, bootstrap, empirical processes

intermediate

Order Statistics & Quantiles

intermediate

Kernel Density Estimation

Bandwidth selection, bias-variance for density estimation

intermediate

The Bootstrap

Efron's bootstrap, bootstrap confidence intervals, bootstrap hypothesis tests

advanced

Empirical Processes & Uniform Convergence

Glivenko-Cantelli, Donsker's theorem, VC dimension

Time-Series & State-Space Methods

Hidden Markov models, state-space inference, and the foundations of time-series statistics