Probabilistic programming language for Bayesian inference
Stan is aprobabilistic programming language forstatistical inference written inC++ .[ 2] statistical model with animperative program calculating the logprobability density function .[ 2]
Stan is licensed under theNew BSD License . Stan is named in honour ofStanislaw Ulam , pioneer of theMonte Carlo method .[ 2]
Stan was created by a development team consisting of 52 members[ 3] Andrew Gelman , Bob Carpenter, Daniel Lee, Ben Goodrich, and others.
A simple linear regression model can be described asy n = α + β x n + ϵ n {\displaystyle y_{n}=\alpha +\beta x_{n}+\epsilon _{n}} ϵ n ∼ normal ( 0 , σ ) {\displaystyle \epsilon _{n}\sim {\text{normal}}(0,\sigma )} y n ∼ normal ( α + β X n , σ ) {\displaystyle y_{n}\sim {\text{normal}}(\alpha +\beta X_{n},\sigma )}
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:
In addition, higher-level interfaces are provided with packages using Stan as backend, primarily in theR language :[ 4]
rstanarm provides a drop-in replacement for frequentist models provided by base R andlme4 using the R formula syntax;brms [ 5] prophet provides automated procedures fortime series forecasting .Stan implements gradient-basedMarkov chain Monte Carlo (MCMC) algorithms for Bayesian inference, stochastic, gradient-basedvariational Bayesian methods for approximate Bayesian inference, and gradient-basedoptimization for penalized maximum likelihood estimation.
MCMC algorithms: Variational inference algorithms:Automatic Differentiation Variational Inference[ 7] Pathfinder: Parallel quasi-Newton variational inference[ 8] Optimization algorithms: Automatic differentiation [ edit ] Stan implements reverse-modeautomatic differentiation to calculate gradients of the model, which is required by HMC, NUTS, L-BFGS, BFGS, and variational inference.[ 2]
Stan is used in fields including social science,[ 9] pharmaceutical statistics ,[ 10] market research ,[ 11] medical imaging .[ 12]
PyMC is a probabilistic programming language in PythonArviZ a Python library for Exploratory Analysis of Bayesian Models^ "Release 2.37.0" . 2 September 2025. Retrieved15 September 2025 .^a b c d e Stan Development Team. 2015.Stan Modeling Language User's Guide and Reference Manual, Version 2.9.0 ^ "Development Team" .stan-dev.github.io . Retrieved2024-11-21 .^ Gabry, Jonah."The current state of the Stan ecosystem in R" .Statistical Modeling, Causal Inference, and Social Science . Retrieved25 August 2020 . ^ "BRMS: Bayesian Regression Models using 'Stan' . 23 August 2021.^ Hoffman, Matthew D.; Gelman, Andrew (April 2014)."The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo" .Journal of Machine Learning Research 15 :pp. 1593–1623. ^ Kucukelbir, Alp; Ranganath, Rajesh; Blei, David M. (June 2015). "Automatic Variational Inference in Stan".1506 (3431).arXiv :1506.03431 Bibcode :2015arXiv150603431K . {{cite journal }}:Cite journal requires|journal= (help ) ^ Zhang, Lu; Carpenter, Bob; Gelman, Andrew; Vehtari, Aki (2022). "Pathfinder: Parallel quasi-Newton variational inference".Journal of Machine Learning Research 23 (306):1– 49. ^ Goodrich, Benjamin King, Wawro, Gregory and Katznelson, Ira, Designing Quantitative Historical Social Inquiry: An Introduction to Stan (2012). APSA 2012 Annual Meeting Paper. Available atSSRN 2105531 ^ Natanegara, Fanni; Neuenschwander, Beat; Seaman, John W.; Kinnersley, Nelson; Heilmann, Cory R.; Ohlssen, David; Rochester, George (2013). "The current state of Bayesian methods in medical product development: survey results and recommendations from the DIA Bayesian Scientific Working Group".Pharmaceutical Statistics .13 (1):3– 12.doi :10.1002/pst.1595 .ISSN 1539-1612 .PMID 24027093 .S2CID 19738522 . ^ Feit, Elea (15 May 2017)."Using Stan to Estimate Hierarchical Bayes Models" . Retrieved19 March 2019 . ^ Gordon, GSD; Joseph, J; Alcolea, MP; Sawyer, T; Macfaden, AJ; Williams, C; Fitzpatrick, CRM; Jones, PH; di Pietro, M; Fitzgerald, RC; Wilkinson, TD; Bohndiek, SE (2019)."Quantitative phase and polarization imaging through an optical fiber applied to detection of early esophageal tumorigenesis" .Journal of Biomedical Optics .24 (12):1– 13.arXiv :1811.03977 Bibcode :2019JBO....24l6004G .doi :10.1117/1.JBO.24.12.126004 .PMC 7006047 PMID 31840442 . 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 theWayback Machine , Proceedings of the NIPS Workshop on Probabilistic Programming.
_logo.png){kind=logo}
