Cited by 7,754Markov Chains and Monte Carlo MethodsData Analysis with RStatistical Methods and Bayesian Inference

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Stan : A Probabilistic Programming Language

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Carpenter, Bob, Gelman, Andrew, Hoffman, Matthew D., Lee, Daniel, Goodrich, Ben, Betancourt, Michael, Brubaker, Marcus, Guo, Jiqiang, Li, Peter, Riddell, Allen·2017·Journal of Statistical Software

DOI 10.18637/jss.v076.i01PMC PMC9788645

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Abstract

Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propagation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces support sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting.

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A Choice‐Based Measure of Issue Importance in the Electorate
Hanretty, C., Lauderdale, B. E., & Vivyan, N. (2020) · American Journal of Political Science · doi:10.1111/ajps.12470 · Cited by 70
“We estimate the conjoint response model by Bayesian posterior simulation, implemented in Stan (Carpenter et al., 2016), using uniform priors on all parameters and reporting posterior means and 95% central intervals.”
Method
The effects of forest cover and disturbance on torrential hazards: large-scale evidence from the Eastern Alps
Sebald, J., Senf, C., Heiser, M., Scheidl, C., Dirk Pflugmacher, A., & Seidl, R. (2019) · Environmental Research Letters · doi:10.1088/1748-9326/ab4937 · Cited by 72
“Joint posterior distributions were sampled using MonteCarlo-Markow-Chain (MCMC) methods implemented in the Software Stan (Carpenter et al 2017) via the rstanarm package (Stan Development Team 2016).”
Method
Tactical departures and strategic arrivals: Divergent effects of climate and weather on caribou spring migrations
Gurarie, E. D., Hebblewhite, M., Joly, K., Kelly, A., Adamczewski, J., Davidson, S. C., Davison, T., Gunn, A., Suitor, M. J., Fagan, W. F., et al. (2019) · Ecosphere · doi:10.1002/ecs2.2971 · Cited by 72
“…model using Bayesian Markov chain Monte Carlo (MCMC), using a Hamiltonian No-U Turn Sampler (NUTS) as implemented in the STAN programming language (Carpenter et al. 2017) via the R package rstan (Stan Development Team 2018) (complete STAN code and implementation in R are provided in Appendix S1).”
Method
Does E-Voting matter for turnout, and to whom?
Petitpas, A., Jaquet, J. M., & Sciarini, P. (2020) · Electoral Studies · doi:10.1016/j.electstud.2020.102245 · Cited by 73
The GALAH survey: multiple stars and our Galaxy
Traven, G., Feltzing, S., Merle, T., der Swaelmen, M. V., Čotar, K., Church, R. P., Zwitter, T., Ting, Y.-S., Sahlholdt, C., Asplund, M., et al. (2020) · Astronomy and Astrophysics · doi:10.1051/0004-6361/202037484 · Cited by 70
The Convergence of Markov Chain Monte Carlo Methods: From the Metropolis Method to Hamiltonian Monte Carlo
Betancourt, M. (2018) · Annalen der Physik · doi:10.1002/andp.201700214 · Cited by 70
“onte Carlo to achieve high performance in a given problem (Hoﬀman and Gelman, 2014) that the method really began to ﬂourish in practice. Along with a user-friendly probabilistic programming language (Carpenter et al., 2016) and a high-performance automatic diﬀerentiation package (Carpenter et al., 2015), the No-U-Turn…”
MethodSelf-citation
The influence of 15-week exercise training on dietary patterns among young adults
Joo, J., Williamson, S. A., Vazquez, A. I., Fernández, J. R., & Bray, M. S. (2019) · International Journal of Obesity · doi:10.1038/s41366-018-0299-3 · Cited by 72
Evaluating Probabilistic Forecasts with scoringRulesPreprint
Jordan, A. I., Krüger, F., & Lerch, S. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1709.04743 · Cited by 70
“MCMC methods are popular across the disciplines, and many sophisticated R implementations are available (e.g., Kastner 2016; Carpenter et al. 2017).”
Method
Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in European countries: technical description updatePreprint
Flaxman, S. R., Mishra, S., Gandy, A., et al. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2004.11342 · Cited by 70
“Fitting was done in the probabilistic programming language Stan[1] using an adaptive Hamiltonian Monte Carlo (HMC) sampler.”
Method
What Kind of Language Is Hard to Language-Model?
Mielke, S. J., Cotterell, R., Gorman, K., Roark, B., & Eisner, J. (2019) · Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics · doi:10.18653/v1/p19-1491 · Cited by 18
“Figure 6 shows the log-probability of held-out data under the regression model, by ﬁxing the estimated difﬁculties d j (and sometimes also the estimated variance σ 2 ) to their values obtained from training data, and then ﬁnding either MAP estimates or posterior means (by running HMC using STAN) of 29 However, version…”
MethodInfluential
Broken into Pieces: ATLAS and Aliqa Uma as One Single Stream
Li, T. S., Koposov, S. E., Erkal, D., Ji, A., Shipp, N., Pace, A., Hilmi, T., Kuehn, K., Lewis, G., Mackey, D., et al. (2020) · Astrophysical Journal · doi:10.3847/1538-4357/abeb18 · Cited by 57
“Software: numpy (van der Walt et al. 2011), scipy (Jones et al. 2001), matplotlib (Hunter 2007), astropy (Astropy Collaboration et al. 2013; Price-Whelan et al. 2018), emcee (Foreman-Mackey et al. 2013), CarPy (Kelson 2003), MOOG (Sneden 1973; Sobeck et al. 2011), smhr (Casey 2014), q3c (Koposov & Bartunov 2006), RVSp…”
Method
Algorithmic risk assessments can alter human decision-making processes in high-stakes government contexts
Green, B., & Chen, Y. (2021) · Proceedings of the ACM on Human-Computer Interaction · doi:10.1145/3479562 · Cited by 69
“ions were almost identical). We implemented models with the brms package in R (16), which provides a high-level interface to Markov Chain Monte Carlo (MCMC) sampling for Bayesian inference using Stan (17). In both settings we regressed the average prediction about each subject (both with and without the risk assessmen…”
Method
Supply chain mapping: a proposed construct
Mubarik, M. S., Kusi‐Sarpong, S., Govindan, K., Khan, S. A., & Oyedijo, A. (2021) · International Journal of Production Research · doi:10.1080/00207543.2021.1944390 · Cited by 70
Action video game play facilitates “learning to learn”
Zhang, R., Chopin, A., Shibata, K., Lu, Z., Jaeggi, S. M., Buschkuehl, M., Green, C. S., & Bavelier, D. (2021) · Communications Biology · doi:10.1038/s42003-021-02652-7 · Cited by 76
Development of a multi-route physiologically based pharmacokinetic (PBPK) model for nanomaterials: a comparison between a traditional versus a new route-specific approach using gold nanoparticles in rats
Chou, W., Cheng, Y., Riviere, J. E., Monteiro-Riviere, N. A., Kreyling, W. G., & Lin, Z. (2022) · Particle and Fibre Toxicology · doi:10.1186/s12989-022-00489-4 · Cited by 71
Synergistic Impacts of Organic Acids and pH on Growth of Pseudomonas aeruginosa: A Comparison of Parametric and Bayesian Non-parametric Methods to Model Growth
Bushell, F. M. L., Tonner, P. D., Jabbari, S., Schmid, A., & Lund, P. A. (2019) · Frontiers in Microbiology · doi:10.3389/fmicb.2018.03196 · Cited by 69
“Model Implementation phenom is implemented using the STAN library, which provides efficient Hamiltonian Monte-Carlo sampling of the model posterior (Carpenter et al., 2017).”
Method
Land‐use change increases climatic vulnerability of migratory birds: Insights from integrated population modelling
Zhao, Q., Arnold, T. W., Devries, J. H., Howerter, D. W., Clark, R. G., & Weegman, M. D. (2019) · Journal of Animal Ecology · doi:10.1111/1365-2656.13043 · Cited by 68
“We implemented the IPM in a hierarchical Bayesian framework using the software stan 2.17.3 (Carpenter et al., 2017), which is called from Program r 3.5.0 (R Development Core Team 2013) through r package “rstan” (Stan Development Team, 2014).”
MethodInfluential
The Neural Basis of Approach-Avoidance Conflict: A Model Based Analysis
Zorowitz, S., Rockhill, A. P., Ellard, K. K., Link, K. E., Herrington, T. M., Pizzagalli, D. A., Widge, A. S., Deckersbach, T., & Dougherty, D. D. (2019) · eNeuro · doi:10.1523/eneuro.0115-19.2019 · Cited by 67
“15 (Carpenter et al., 2017) with four chains of 2000 steps each (1000 burn-in, thinning 4), yielding 1000 posterior samples total.”
Background
Probabilistic human health risk assessment of perfluorooctane sulfonate (PFOS) by integrating in vitro, in vivo toxicity, and human epidemiological studies using a Bayesian-based dose-response assessment coupled with physiologically based pharmacokinetic (PBPK) modeling approach
Chou, W., & Lin, Z. (2020) · Environment International · doi:10.1016/j.envint.2020.105581 · Cited by 69
Investigating occupant injury severity of truck-involved crashes based on vehicle types on a mountainous freeway: A hierarchical Bayesian random intercept approach
Haq, M. T., Zlatkovic, M., & Ksaibati, K. (2020) · Accident Analysis & Prevention · doi:10.1016/j.aap.2020.105654 · Cited by 68
Model-based evaluation of school- and non-school-related measures to control the COVID-19 pandemic
Rozhnova, G., van Dorp, C. H., Bruijning-Verhagen, P. C. J. L., Bootsma, M., van de Wijgert, J. H. H. M., Bonten, M. J. M., & Kretzschmar, M. (2021) · Nature Communications · doi:10.1038/s41467-021-21899-6 · Cited by 71
The epidemiology of Mayaro virus in the Americas: A systematic review and key parameter estimates for outbreak modelling
Caicedo, E.-Y., Charniga, K., Rueda-Monsalbe, A., Dorigatti, I., Méndez, Y. R., Hamlet, A., Carrera, J., & Cucunubá, Z. M. (2021) · PLoS neglected tropical diseases · doi:10.1371/journal.pntd.0009418 · Cited by 68
“We run 10,000 iterations with 5,000 burn-in and 138 selected the models after convergence and based on the expected log predictive density for an 139 out-of-sample data point (elpd) [29, 30].”
Method
The Photon Ring in M87*
Broderick, A. E., Pesce, D. W., Gold, R., Tiede, P., Pu, H.-Y., Anantua, R., Britzen, S., Ceccobello, C., Chatterjee, K., Cheng, X., et al. (2022) · The Astrophysical Journal · doi:10.3847/1538-4357/ac7c1d · Cited by 69
“In summary, the improved sampler makes use of a deterministic even-odd swap tempering scheme (Syed et al. 2019) using the Hamiltonian Monte Carlo sampling kernel from the Stan package (Carpenter et al. 2017).”
Method
A resource-rational model of human processing of recursive linguistic structure
Hahn, M., Futrell, R., Lévy, R., & Gibson, E. A. F. (2022) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2122602119 · Cited by 80
Linking changes in species composition and biomass in a globally distributed grassland experiment
Ladouceur, E. R., Blowes, S. A., Chase, J. M., Clark, A. T., Garbowski, M., Alberti, J., Arnillas, C. A., Bakker, J. D., Barrio, I. C., Bharath, S., et al. (2022) · Ecology Letters · doi:10.1111/ele.14126 · Cited by 68
“For Bayesian inferences and estimates of uncertainty, all models described above were fitted using the Hamiltonian Monte Carlo (HMC) sampler Stan (Carpenter et al., 2017) and coded using the ‘brms’ package (Bürkner, 2018) in the R for Statistical Computing and Graphics environment (v.4.0.2; (R Core Development Team, 2…”
Influential
Fast Bayesian Inference of Reparameterized Gamma Process With Random Effects
Zhou, S., Xu, A., Tang, Y., et al. (2023) · IEEE Transactions on Reliability · doi:10.1109/tr.2023.3263940 · Cited by 68
“Although Stan [52] integrates the VB algorithm for engineering applications, it directly approximates the model parameters of immediate interest with a multivariate normal distribution.”
Method
High-Dimensional Bayesian Geostatistics
Banerjee, S. (2017) · Bayesian Analysis · doi:10.1214/17-ba1056r · Cited by 67
Neural mechanisms underlying the effects of physical fatigue on effort-based choice
Hogan, P. S., Chen, S., Teh, W. W., & Chib, V. S. (2020) · Nature Communications · doi:10.1038/s41467-020-17855-5 · Cited by 83
A synthesis of land use impacts on stream biodiversity across metrics and scales
Petsch, D. K., Blowes, S. A., MELO, A. S., & Chase, J. M. (2021) · Ecology · doi:10.1002/ecy.3498 · Cited by 69
AI makes you smarter but none the wiser: The disconnect between performance and metacognition
Fernandes, D., Villa, S., Nicholls, S., Haavisto, O., Buschek, D., Schmidt, A., Kosch, T., Shen, C., & Welsch, R. (2025) · Computers in Human Behavior · doi:10.1016/j.chb.2025.108779 · Cited by 144
Temporal probabilistic modeling of bacterial compositions derived from 16S rRNA sequencing
Äijö, T., Müller, C. L., & Bonneau, R. (2017) · Bioinformatics · doi:10.1093/bioinformatics/btx549 · Cited by 64
“In light of these limitations, several methodologies have been proposed for simultaneous modeling of taxa through their relative abundances, such as the Dirichlet-multinomial (DM) (Holmes 2012; Chen and Li, 2013) and logistic normal multinomial models (Xia 2013).”
Method
Continuous Monitoring of A/B Tests without Pain: Optional Stopping in Bayesian Testing
Deng, A., Lu, J., & Chen, S. (2016) · IEEE International Conference on Data Science and Advanced Analytics (DSAA) · doi:10.1109/dsaa.2016.33 · Cited by 67
Bat genomes illuminate adaptations to viral tolerance and disease resistance
Morales, A. E., Dong, Y., Brown, T., Baid, K., Kontopoulos, D. G., Gonzalez, V., Huang, Z., Ahmed, A.-W., Bhuinya, A., Hilgers, L., et al. (2025) · Nature · doi:10.1038/s41586-024-08471-0 · Cited by 78
Value and limitations of intracranial recordings for validating electric field modeling for transcranial brain stimulation
Puonti, O., Saturnino, G. B., Madsen, K. H., & Thielscher, A. (2019) · NeuroImage · doi:10.1016/j.neuroimage.2019.116431 · Cited by 66
“The Stan code for running the analysis is provided in Supplementary Material 1.”

Authors: Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., & Riddell, A.
Affiliations: Columbia University (US)
Adobe Systems (United States) (US)
York University (CA)
Indiana University (US)
ORCID: Carpenter, Bob
Gelman, Andrew
Lee, Daniel
Betancourt, Michael
Brubaker, Marcus
Li, Peter
Riddell, Allen
Venue: Journal of Statistical Software
Publisher: Foundation for Open Access Statistic
Published: 2017
Volume: 76
Issue: 1
Pages: 1-32
ISSN: 1548-7660 (ISSN-L)
DOI: 10.18637/jss.v076.i01
PMC: PMC9788645
OpenAlex: W2577537660
Semantic Scholar: f8110d834fcc9a3a1c4416d4b165984219fe26f4
Cited by: 7,754
References: 49
Citations by year: 2026: 672 · 2025: 811 · 2024: 811 · 2023: 835 · 2022: 917 · 2021: 1127 · 2020: 970 · 2019: 752 · 2018: 516 · 2017: 253
FWCI: 474.30
Type: journal article
Language: en
Open access: diamond
doi.org
Licence: cc-by
Locations: Journal of Statistical Software (publishedVersion, cc-by)
PubMed (publishedVersion)
PubMed Central (acceptedVersion)
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) (submittedVersion, other-oa)
RePEc: Research Papers in Economics (submittedVersion)
DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
Journal of Systems and Software
Grantee Submission
Funders: National Science Foundation (0934516, 1205516)
U.S. Department of Energy (DESC0002099)
Harvard University
National Institutes of Health (1g20rr030893-01)
Institute of Education Sciences (R305D090006-09A, R305D090006)
National Center for Research Resources (G20RR030893)
Topics: Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
Physical Sciences › Mathematics › Statistics and Probability › Data Analysis with R
Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
Keywords: Stan, probabilistic programming languages, Bayesian inference, Markov chain Monte Carlo, No-U-Turn Sampler, Hamiltonian Monte Carlo, L-BFGS optimization, penalized likelihood, variational inference, expectation propagation, marginal inference, R interface, Python interface, posterior analysis

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Stan : A Probabilistic Programming Language

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