Stan (software)

Stan
Original author	Stan Development Team
Initial release	August 30, 2012 (2012-08-30)
Stable release	2.39.01  / 19 May 2026 (19 May 2026)
Written in	C++
Operating system	Unix-like, Microsoft Windows, Mac OS X
Platform	Intel x86 - 32-bit, x64
Type	Statistical package
License	New BSD License
Website	mc-stan.org
Repository	github.com/stan-dev/stan

Stan is a probabilistic programming language for statistical inference written in C++.² The Stan language is used to specify a (Bayesian) statistical model with an imperative program calculating the log probability density function.²

Stan is licensed under the New BSD License. Stan is named in honour of Stanislaw Ulam, pioneer of the Monte Carlo method.²

Stan was created by a development team consisting of 52 members³ that includes Andrew Gelman, Bob Carpenter, Daniel Lee, Ben Goodrich, and others.

A simple linear regression model can be described as ${\displaystyle y_{n}=\alpha +\beta x_{n}+\epsilon _{n}}$, where ${\displaystyle \epsilon _{n}\sim {\text{normal}}(0,\sigma )}$. This can also be expressed as ${\displaystyle y_{n}\sim {\text{normal}}(\alpha +\beta X_{n},\sigma )}$. The latter form can be written in Stan as the following:

data {
  int<lower=0> N;
  vector[N] x;
  vector[N] y;
}
parameters {
  real alpha;
  real beta;
  real<lower=0> sigma;
}
model {
  y ~ normal(alpha + beta * x, sigma);
}

The Stan language itself can be accessed through several interfaces:

• CmdStan – a command-line executable for the shell,
• CmdStanR and rstan – R software libraries,
• CmdStanPy and PyStan – libraries for the Python programming language,
• CmdStan.rb - library for the Ruby programming language,
• MatlabStan – integration with the MATLAB numerical computing environment,
• Stan.jl – integration with the Julia programming language,
• StataStan – integration with Stata.
• Stan Playground - online at [1]

In addition, higher-level interfaces are provided with packages using Stan as backend, primarily in the R language:⁴

• rstanarm provides a drop-in replacement for frequentist models provided by base R and lme4 using the R formula syntax;
• brms⁵ provides a wide array of linear and nonlinear models using the R formula syntax;
• prophet provides automated procedures for time series forecasting.

Stan implements gradient-based Markov chain Monte Carlo (MCMC) algorithms for Bayesian inference, stochastic, gradient-based variational Bayesian methods for approximate Bayesian inference, and gradient-based optimization for penalized maximum likelihood estimation.

• MCMC algorithms:
  • Hamiltonian Monte Carlo (HMC)
  • No-U-Turn sampler²⁶ (NUTS), a variant of HMC and Stan's default MCMC engine
• Variational inference algorithms:
  • Automatic Differentiation Variational Inference⁷
  • Pathfinder: Parallel quasi-Newton variational inference⁸
• Optimization algorithms:
  • Limited-memory BFGS (L-BFGS) (Stan's default optimization algorithm)
  • Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS)
  • Laplace's approximation for classical standard error estimates and approximate Bayesian posteriors

Stan implements reverse-mode automatic differentiation to calculate gradients of the model, which is required by HMC, NUTS, L-BFGS, BFGS, and variational inference.² The automatic differentiation within Stan can be used outside of the probabilistic programming language.

Stan is used in fields including social science,⁹ pharmaceutical statistics,¹⁰ market research,¹¹ and medical imaging.¹²