Stan (software)

Stan

Original author(s)	Stan Development Team
Initial release	August 30, 2012 (2012-08-30)
Stable release	2.35.0[1]  / 3 June 2024; 6 months ago (3 June 2024)
Repository	github.com/stan-dev/stan
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

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 members^[3] 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;
• brms^[5] 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 sampler^[2][6] (NUTS), a variant of HMC and Stan's default MCMC engine
• Variational inference algorithms:
  • Automatic Differentiation Variational Inference^[7]
  • Pathfinder: Parallel quasi-Newton variational inference^[8]
• Optimization algorithms:
  • Limited-memory BFGS (Stan's default optimization algorithm)
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
  • 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]

• PyMC is a probabilistic programming language in Python
• ArviZ a Python library for Exploratory Analysis of Bayesian Models

• Carpenter, Bob; Gelman, Andrew; Hoffman, Matthew; Lee, Daniel; Goodrich, Ben; Betancourt, Michael; Brubaker, Marcus; Guo, Jiqiang; Li, Peter; Riddell, Allen (2017). "Stan: A Probabilistic Programming Language". Journal of Statistical Software. 76 (1): 1–32. doi:10.18637/jss.v076.i01. ISSN 1548-7660. PMC 9788645. PMID 36568334.
• Gelman, Andrew, Daniel Lee, and Jiqiang Guo (2015). Stan: A probabilistic programming language for Bayesian inference and optimization, Journal of Educational and Behavioral Statistics.
• Hoffman, Matthew D., Bob Carpenter, and Andrew Gelman (2012). Stan, scalable software for Bayesian modeling Archived 2015-01-21 at the Wayback Machine, Proceedings of the NIPS Workshop on Probabilistic Programming.

• Stan web site
• Stan source, a Git repository hosted on GitHub