Lean 4 Formalization of Statistical Theory

Website

https://statlean.github.io/website/

The interactive website lets you browse every formalized result and compare its informal mathematical statement side-by-side with the Lean 4 proof signature.

📐 Informal ↔ Lean side-by-side Natural-language statement with LaTeX math and textbook citation alongside the exact Lean theorem or def signature.	🔦 Hypothesis cross-highlighting Hover any hypothesis in the Lean code to highlight the corresponding phrase in the informal statement, and vice versa.
🕸️ Dependency graphs Every result has an interactive graph showing the chain of repository lemmas down to the first Mathlib boundary, color-coded by topic.	📝 Note on Informalization Each result includes a formalization note explaining design choices, typeclass decisions, and how the Lean encoding relates to the textbook statement. Links to the doc-gen4 API reference for every result.

_{Informal statement (left) aligned with Lean signature (right) — hover a hypothesis to cross-highlight.}

_{Global dependency graph — color-coded by topic, click any node to open the result.}

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Installation Guide

1. Install Lean via elan

elan manages Lean versions and reads the lean-toolchain file to pin the exact version used here.

macOS / Linux / WSL:

curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh -s -- -y
source $HOME/.elan/env   # add to ~/.zshrc or ~/.bashrc

Windows (PowerShell):

curl -L https://github.com/leanprover/elan/releases/latest/download/elan-init.ps1 -o elan-init.ps1
.\elan-init.ps1

Verify:

elan --version
lean --version    # may say "no toolchain" until step 3 — that is fine
lake --version

2. Clone

git clone https://github.com/StatLean/Stat-Lean.git
cd Stat-Lean

The library is the single lean_lib StatLean, organized into per-area sublibraries under StatLean/:

• StatLean/AsymptoticStatistics/ — parametric & semiparametric asymptotics (van der Vaart, Asymptotic Statistics)
• StatLean/Bayesian/ — posteriors via disintegration, conjugacy, Bayesian decision theory, hierarchical & empirical Bayes, MCMC correctness, posterior contraction (Robert, The Bayesian Choice; Bhattacharya–Pati–Pillai–Dunson)
• StatLean/CausalInference/ — potential-outcomes causal inference: randomized experiments (Fisher's randomization test, Neyman's variance, stratified & matched-pair designs), observational identification (unconfoundedness, standardization, propensity score, IPW, doubly robust AIPW, ATT, matching & trimming), sensitivity analysis (Manski bounds, E-value, Rosenbaum) and instrumental variables (LATE/CACE, instrumental inequalities, linear IV) (Ding, A First Course in Causal Inference; Imbens & Rubin, Causal Inference for Statistics, Social, and Biomedical Sciences)
• StatLean/ConcentrationInequalities/ — sub-Gaussian / sub-exponential / Bernstein / maximal inequalities and empirical processes (Lu, Big Data Analysis)
• StatLean/HighDimensionalStatistics/ — OLS, Lasso rates, compressed sensing, M-estimators (Lu, Big Data Analysis; Wainwright)
• StatLean/MultipleTesting/ — FDR / FWER control, knockoffs, goodness-of-fit (Lu, Big Data Analysis; Candès)
• StatLean/Minimaxity/ — Le Cam and Fano minimax lower bounds (Wainwright)
• StatLean/Optimization/ — gradient, proximal, and accelerated methods (Lu, Big Data Analysis)
• StatLean/StatisticalModels/ — compositional statistical models: model calculus & identifiability, survival / event history, Gaussian conditioning substrate, mixed effects, longitudinal / GEE, missing data (Andersen–Borgan–Gill–Keiding; Liang–Zeger; Rubin)

3. Fetch the Mathlib build cache

lake exe cache get

Mathlib is large; this downloads a precompiled cache in minutes instead of building from scratch.

4. Build

lake build

5. Use as a dependency

Add to your lakefile.lean:

require StatLean from git
  "https://github.com/StatLean/Stat-Lean.git" @ "main"

then run lake update && lake exe cache get && lake build. Your project must use the same toolchain version as this repository's lean-toolchain file.

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Topics

Browse the library by area on the website:

• Bayesian Statistics — priors and posteriors via disintegration, sufficiency and conjugate families, Bayesian decision theory and the Bayes route to minimaxity, hierarchical and empirical Bayes, MCMC correctness, and the Bernstein–von Mises theorem.
• Causal Inference — potential outcomes and estimands, randomized experiments and Neyman inference, identification by adjustment and matching, augmented inverse-probability weighting and double robustness, and instrumental variables.
• Computational Statistics — Monte Carlo estimation and its error, importance sampling and the optimal importance density, rejection sampling, multinomial resampling of particles, the finite-sample bootstrap and jackknife, and cross-validation.
• Probability Inequalities — sub-Gaussian and sub-exponential tails, Hoeffding, Bernstein and McDiarmid inequalities, maximal inequalities and chaining over covering classes, and the empirical-process limits built on them.
• Experimental Design — randomization and sampling designs as distributions over allocations, completely randomized and blocked experiments, the analysis-of-variance identity, factorial characters, and Horvitz–Thompson estimation.
• High-Dimensional Statistics — OLS mean-squared error, deterministic and random-noise Lasso rates, support recovery, compressed-sensing recovery, and M-estimator deviation bounds.
• Hypothesis Tests — the Neyman–Pearson lemma, monotone likelihood ratio and UMP tests, unbiasedness and Neyman structure, invariance and maximal invariants, goodness-of-fit, permutation tests and the bootstrap.
• Minimaxity — the estimation-to-testing reduction, Le Cam's two-point and convex-hull methods, Fano's inequality with local packing, and worked minimax rates.
• Miscellaneous Results — load-bearing probability and analysis: optional stopping, order statistics, divergences and entropy, packing bounds, and classical limit theorems.
• Multiple Testing — Benjamini–Hochberg FDR control, Holm/Bonferroni FWER control, the knockoff filter, e-values, conformal coverage, and goodness-of-fit tests.
• Nonparametric Statistics — kernel density estimation over Hölder classes, local polynomial regression, projection estimators, and reproducing-kernel Hilbert spaces from Moore–Aronszajn to Mercer's theorem.
• Optimization — convexity and smoothness primitives, and the convergence rates of gradient descent, Frank–Wolfe, proximal, and Nesterov-accelerated methods.
• Parametric Statistics — the delta method; moment, M-, Z-, and maximum-likelihood estimation; differentiability in quadratic mean; the LAN expansion; and the Hájek–Le Cam convolution and local asymptotic minimax bounds.
• Point Estimation — exponential families and sufficiency, completeness and Basu's theorem, the Rao–Blackwell and Lehmann–Scheffé route to UMVU estimators, the Cramér–Rao inequality, and equivariant estimation.
• Robust Statistics — gross-error contamination, influence functions and breakdown points, Huber M-estimation and the median's minimax bias, scale and regression quantiles, and sub-Gaussian mean estimation under heavy tails.
• Semiparametric Statistics — tangent spaces, efficient influence functions, score operators, and the semiparametric convolution and minimax bounds.
• Statistical Learning — empirical risk minimization and uniform convergence, PAC and agnostic learnability, VC dimension and growth functions, Rademacher complexity and symmetrization, stability and PAC-Bayes bounds.
• Statistical Models — survival analysis with censoring, Kaplan–Meier and the Cox model, linear mixed effects and BLUP, GEE sandwich covariance, missing-data mechanisms with IPW, factor models, and graphical models with d-separation.
• Time Series — stationarity and autocovariance, the Herglotz representation and spectral density, linear filters and ARMA, strong-mixing coefficients and their limit theorems, and the ARCH, GARCH and volatility models.

Each topic page aligns every result's informal statement with its Lean 4 signature, alongside a reference block and an interactive dependency graph.

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References

A full bibliography — every textbook and paper cited across the library, together with the results that cite each one — lives on the website:

Reference page → https://statlean.github.io/website/#/references