Stan Bayesian Inference Engine

Stan is a probabilistic programming language and statistical inference engine for Bayesian modeling, providing full Bayesian inference, approximate Bayesian inference, and penalized maximum likelihood estimation. Named after Stanislaw Ulam (Monte Carlo pioneer), Stan was created by Andrew Gelman and Bob Carpenter at Columbia University in 2012. Key features: Hamiltonian Monte Carlo (HMC) and the No-U-Turn Sampler (NUTS) for efficient full Bayesian inference, auto-tuned during warmup to adapt step size and mass matrix, dramatically reducing manual tuning compared to Metropolis-Hastings and Gibbs sampling. Variational inference: Automatic Differentiation Variational Inference (ADVI) for approximate posterians when full MCMC is too slow, providing orders-of-magnitude speedup. Penalized MLE via L-BFGS with automatic differentiation for point estimates. Probabilistic programming language with blocks (data, transformed data, parameters, model, generated quantities) for specifying arbitrary Bayesian models including hierarchical, mixture, time series, spatial, and ODE-based models. Automatic differentiation: reverse-mode autodiff in C++ supporting over 40 probability distributions (normal, Cauchy, Student-t, beta, gamma, Dirichlet, multinomial, Poisson, negative binomial). Interfaces: RStan (R), PyStan (Python), CmdStan (CLI), MatlabStan, Stan.jl. Diagnostics: R-hat convergence, effective sample size (ESS), divergent transitions, energy diagnostics. Posterior predictive checking and model comparison via WAIC and LOO-CV. Cross-platform: Linux, macOS, Windows. Open source under BSD-3-Clause.