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Stan : A Probabilistic Programming Language
Open accessAbstract
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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“When dealing with more complex models that go beyond the staple analyses such as t tests, there exist a number of software packages that allow custom coding, such as JAGS (Plummer, 2003) or Stan (Carpenter et al., 2017).”
From The JASP guidelines for conducting and reporting a Bayesian analysis (2021) · cites this paper as Carpenter et al., 2017
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“Stan code of the hierarchical linear regression and ordinal regression models and details on choice of model priors.”
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“All models were implemented and fit using the Stan modeling language (Carpenter et al., 2017), which performs Bayesian inference using Hamiltonian Monte Carlo sampling.”
MethodProbabilistic Inference on Noisy Time Series (PINTS)
Clerx, M., Robinson, M. J., Lambert, B. C., et al. (2019) · Journal of Open Research Software · doi:10.5334/jors.252 · Cited by 64
“e model, useful for very expensive model evaluations. Other software packages that enable parameter inference and sampling for ODE models include BioBayes [21], ABC-SysBio [12], SYSBIONS [7] and Stan [2]. These packages use either a common model description format (for example, SBML) or their own language (for example…”
BackgroundMethodNegative emotion boosts quality of visual working memory representation.
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Estimating Smooth Country–Year Panels of Public Opinion
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“The six models are estimated using Bayesian Markov-Chain Monte Carlo (MCMC) methods via Stan software, which implements Hamiltonian Monte Carlo sampling (Carpenter et al. 2017; Stan Development Team 2017).”
MethodInfluentialImpact of Tanzania’s Wildlife Management Areas on household wealth
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Bayesian inference of network structure from unreliable data
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“Empirical findings aside, Cesario et al. (2019) argue convincingly that to understand racial disparities in killings by police, we have to engage in proper benchmarking, such that we compare the relative rates of police use-of-force against criminals, fB fW , and noncriminals, yB yW , independently. The push to take b…”
BackgroundMethodInfluentialVos, E. R. A., van Boven, M., den Hartog, G., Backer, J. A., Klinkenberg, D., van Hagen, C. C. E., Boshuizen, H. C., van Binnendijk, R. S., Mollema, L., van der Klis, F. R. M., et al. (2021) · Clinical Infectious Diseases · doi:10.1093/cid/ciab264 · Cited by 55
Nilsson, A., Suttie, N., Stoner, J. S., & Muscheler, R. (2022) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2200749119 · Cited by 63
Meiotic DNA breaks drive multifaceted mutagenesis in the human germ line
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Wallis, T. S. A., Funke, C. M., Ecker, A. S., Gatys, L. A., Wichmann, F. A., & Bethge, M. (2017) · Journal of Vision · doi:10.1167/17.12.5 · Cited by 67
The Attention Network Test Database: ADHD and Cross-Cultural Applications
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“As in the previous application, quantitative inference was achieved through a Bayesian hierarchical model, coded and sampled using Stan (Carpenter et al., 2017).”
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Global estimates of paediatric tuberculosis incidence in 2013–19: a mathematical modelling analysis
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“We used Hamiltonian Monte Carlo 34 to obtain samples from the posterior distribution (appendix 1 pp 7–8), with the model fit to all calibration data simultaneously.”
Analysis of the European baseline survey of norovirus in oysters
(EFSA), E. F. S. A. (2019) · EFSA Journal · doi:10.2903/j.efsa.2019.5762 · Cited by 62
“Analyses were performed using R-stan (version 2.18.2) (Carpenter et al., 2017; Stan development team, 2018) and brms package (version 2.8.0) (B€urkner, 2017, 2018) in R (version 3.5.3).”
MethodInfluentialThe seasonal sensitivity of brown bear denning phenology in response to climatic variability
Delgado, M. D. M., Tikhonov, G. N., Meyke, E., et al. (2018) · Frontiers in Zoology · doi:10.1186/s12983-018-0286-5 · Cited by 62
“Carpenter B, Gelman A, Hoffman MD, Lee D, Goodrich B, Betancourt M, et al. Stan : a probabilistic programming language.”
MethodInfluentialSmith, K. M., & Apicella, C. L. (2020) · Evolution and Human Behavior · doi:10.1016/j.evolhumbehav.2020.07.009 · Cited by 63
Lionetti, F., Klein, D. N., Pastore, M., Aron, E. N., Aron, A. P., & Pluess, M. (2021) · European Child & Adolescent Psychiatry · doi:10.1007/s00787-021-01830-6 · Cited by 62
Spatial and temporal fluctuations in COVID-19 fatality rates in Brazilian hospitals
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A Research Note on the Prevalence of Housing Eviction Among Children Born in U.S. Cities
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greta: simple and scalable statistical modelling in R
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“This can be seen for example in the development of Stan (Carpenter et al., 2017).”
BackgroundMethodHarrison, J. G., Calder, W. J., Shastry, V., & Buerkle, C. A. (2019) · Molecular Ecology Resources · doi:10.1111/1755-0998.13128 · Cited by 55
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Dogmatism manifests in lowered information search under uncertainty
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“tion over the parameters were extracted using the Python package PyStan (Carpenter et al., 2017).”
MethodInfluentialNiehus, R., Salazar, P. M. D., Taylor, A. R., et al. (2020) · The Lancet Infectious Diseases · doi:10.1016/s1473-3099(20)30229-2 · Cited by 61
Moe, S. J., Wolf, R. L., Xie, L., Landis, W. G., Kotamäki, N., & Tollefsen, K. E. (2020) · Integrated Environmental Assessment and Management · doi:10.1002/ieam.4348 · Cited by 61
“All relationships (Table 2) were fitted to the data using the R package brms (Bürkner 2017), which is a high‐level interface to statistical inference language Stan (Carpenter et al. 2017).”
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Schmied, G., Pretzsch, H., Ambs, D., Uhl, E., Schmucker, J., Fäth, J., Biber, P., Hoffmann, Y.-D., Šeho, M., Mellert, K. H., et al. (2023) · Forest Ecology and Management · doi:10.1016/j.foreco.2023.121305 · Cited by 63
Neanderthal ancestry through time: Insights from genomes of ancient and present-day humans
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Massive data compression for parameter-dependent covariance matrices
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MethodMultiBUGS: A Parallel Implementation of the BUGS Modeling Framework for Faster Bayesian Inference
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“posterior distribution. These ideas have also been widely adopted in other Bayesian software, notably in JAGS (Plummer, 2017) and NIMBLE (de Valpine et al., 2017), and related ideas are used in Stan (Carpenter et al., 2017). Technological advances in recent years have led to massive increases in the amount of data tha…”
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“All preregistered analyses discussed in the following were implemented as scripts in the R programming language (R Core Team, 2015) using the BayesFactor package (Morey & Rouder, 2015) and Stan (Carpenter et al., 2017) as the modelling backend and R- packages rstan (Stan Development Team, 2016) and brms (Bürkner, 2017…”
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“Statistical analyses To estimate the effects of the three inbreeding coefficients FROHlong,FROHmedium and FROHshort on survival, we fitted a binomial Bayesian generalised linear mixed model (GLMM) with logit link, using brms (Bürkner, 2017), a high-level R interface to Stan (Carpenter et al., 2017).”
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“Stan: A Probabilistic Programming Language.”
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“We used the R package brms [38] to fit Bayesian regression models using the probabilistic programming language Stan [39], and adjusted for average paternal age within families to isolate the effect of paternal age differences between siblings.”
Mowinckel, A. M., Alnæs, D., Pedersen, M. L., Ziegler, S., Fredriksen, M., Kaufmann, T., Sonuga-Barke, E. J. S., Endestad, T., Westlye, L. T., & Biele, G. (2017) · NeuroImage Clinical · doi:10.1016/j.nicl.2017.03.008 · Cited by 71
“0-4]) (Carpenter et al., 2017; Stan Development Team, 2016) was used for inference testing on the edges, node variance, and task parameter betas.”
InfluentialThe anisotropy of personal space
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Chen, Y., Armoiry, X., Higenbottam, C. V., Cowley, N. J., Basra, R., Watson, S., Tarrant, C., Boyal, A., Sutton, E., Wu, C., et al. (2019) · BMJ Open · doi:10.1136/bmjopen-2018-025764 · Cited by 68
“16.(22) Four HMC chains were run for 10 000 iterations including 2000 warm-up/burn-in iterations or more iterations in the same proportion if convergence was judged not to have been achieved.”
Using Response Times for Joint Modeling of Response and Omission Behavior
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“Bayesian estimation was conducted using Stan version 2.18 (Carpenter et al., 2017) employing the rstan package (Guo, Gabry, & Goodrich, 2018) for R version 3.51 (R Development Core Team, 2017).”
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“To fit our models to the data, we use the Python implementation of the Stan statistical software package (Carpenter et al. 2017).3 Stan uses the No-U-Turn Sampler (NUTS) Markov Chain Monte Carlo (MCMC) algorithm, a method of numerically evaluating hierarchical Bayesian models.”
MethodNeuronal correlates of strategic cooperation in monkeys
Ong, W. S., Madlon‐Kay, S., & Platt, M. L. (2020) · Nature Neuroscience · doi:10.1038/s41593-020-00746-9 · Cited by 61
Östling, R., Börstell, C., & Courtaux, S. (2018) · Frontiers in Psychology · doi:10.3389/fpsyg.2018.00725 · Cited by 59
“For inference, we used the NUTS sampler implemented in the Stan software package (Carpenter et al., 2017) to run four independent chains with 5,000 burn-in iterations followed by 5,000 sampling iterations.”
MethodMacRitchie, J., Breaden, M., Milne, A. J., & McIntyre, S. (2020) · Frontiers in Psychology · doi:10.3389/fpsyg.2019.02868 · Cited by 68
The response of canopy height diversity to natural disturbances in two temperate forest landscapes
Senf, C., Mori, A., Müller, J. C., & Seidl, R. (2020) · Landscape Ecology · doi:10.1007/s10980-020-01085-7 · Cited by 63
Attenuated Directed Exploration during Reinforcement Learning in Gambling Disorder
Wiehler, A., Chakroun, K., & Peters, J. (2021) · Journal of Neuroscience · doi:10.1523/jneurosci.1607-20.2021 · Cited by 53
Rouy, M., Saliou, P., Nalborczyk, L., Pereira, M., Roux, P., & Faivre, N. (2021) · Neuroscience & Biobehavioral Reviews · doi:10.1016/j.neubiorev.2021.03.017 · Cited by 59
Sociodemographic characteristics of missing data in digital phenotyping
Kiang, M. V., Chen, J. T., Krieger, N., Buckee, C. O., Alexander, M. J., Baker, J. T., Buckner, R. L., Coombs, G., Rich-Edwards, J. W., Carlson, K. W., et al. (2021) · Scientific Reports · doi:10.1038/s41598-021-94516-7 · Cited by 55
Truchado, P., Gil, M. I., López, C., Garre, A., López-Aragón, R. F., Böhme, K., & Allende, A. (2021) · International Journal of Food Microbiology · doi:10.1016/j.ijfoodmicro.2021.109352 · Cited by 59
Quantifying the interplay of conversational devices in building mutual understanding.
Dideriksen, C., Christiansen, M. H., Tylén, K., Dingemanse, M., & Fusaroli, R. (2022) · Journal of Experimental Psychology General · doi:10.1037/xge0001301 · Cited by 72
Sadil, P. S., Cowell, R. A., & Huber, D. E. (2023) · Psychonomic Bulletin & Review · doi:10.3758/s13423-023-02320-3 · Cited by 59
“Models were constructed and fit using the Stan language (Carpenter et al., 2016), using its interface (RStan, 2.18.2) for the R computing language (R Core Team).”
MethodInfluentialDenotational validation of higher-order Bayesian inference
Ścibior, A., Kammar, O., Vákár, M., et al. (2017) · Proceedings of the ACM on Programming Languages · doi:10.1145/3158148 · Cited by 59
“In the past five years, however, designing such a language and building its runtime system has become an active research area, and lead to practical programming languages and libraries [Carpenter et al. 2017; Goodman et al. 2008; Goodman and Stuhlmüller 2014; Gordon et al. 2014; Mansinghka et al. 2014; Minka et al. 20…”
BackgroundOn the kinematics of a runaway Be star population
Boubert, D., & Evans, N. W. (2018) · Monthly Notices of the Royal Astronomical Society · doi:10.1093/mnras/sty980 · Cited by 59
Geographic and Racial Disparities in Infant Hearing Loss
Lantos, P. M., Panayotti, G. M. M., Barber, X., Raynor, E. M., Tucci, D. L., Hoffman, K., Permar, S. R., Jackson, P., Hughes, B. L., Kind, A., et al. (2018) · Otolaryngology · doi:10.1177/0194599818803305 · Cited by 59
Bayesian Data Analysis in Empirical Software Engineering Research
Furia, C. A., FELDT, R. H., & Torkar, R. (2019) · IEEE Transactions on Software Engineering · doi:10.1109/tse.2019.2935974 · Cited by 67
“Tools such as Stan [16] provide scalable generic implementations of these techniques.”
MethodBidirectional cingulate-dependent danger information transfer across rats
Han, Y., Bruls, R., Soyman, E., Thomas, R. M., Pentaraki, V., Jelinek, N., Heinemans, M., Bassez, I., Verschooren, S., Pruis, I., et al. (2019) · PLoS Biology · doi:10.1371/journal.pbio.3000524 · Cited by 55
“Model fitting and parameter estimation were conducted using Bayesian analysis by estimating the posterior distribution through Bayes’ rule using in-house code in R Stan [70] in R version 3.”
MethodInfluentialSolesio, M. E., del Molino, L. C. G., Elustondo, P. A., Diao, C. T. M., Chang, J., & Pavlov, E. V. (2019) · Cell Calcium · doi:10.1016/j.ceca.2019.102127 · Cited by 59
Allometry and Ecology of the Bilaterian Gut Microbiome
Sherrill-Mix, S. A., McCormick, K., Lauder, A. P., Bailey, A., Zimmerman, L., Li, Y., Django, J.-B. N., Bertolani, P., Colin, C., Hart, J. A. L., et al. (2018) · mBio · doi:10.1128/mbio.00319-18 · Cited by 61
“To estimate the species-area relationship in animal guts, fit a Bayesian regression model of the rarefied OTU counts observed in each animal gut in relation to the mean weight of the hosts using Stan (72).”
Pizo, M. A., Morales, J. M., Ovaskainen, O., & Carlo, T. A. (2020) · The American Naturalist · doi:10.1086/712381 · Cited by 62
Hydrologic multi-model ensemble predictions using variational Bayesian deep learning
Li, D., Marshall, L. A., Liang, Z., & Sharma, A. (2021) · Journal of Hydrology · doi:10.1016/j.jhydrol.2021.127221 · Cited by 63
Progression subtypes in Parkinson’s disease identified by a data-driven multi cohort analysis
Hähnel, T., Raschka, T., Sapienza, S., Klucken, J., Glaab, E., Corvol, J.-C., Falkenburger, B. H., & Fröhlich, H. (2024) · npj Parkinson's Disease · doi:10.1038/s41531-024-00712-3 · Cited by 58
Effects of intranasal oxytocin on symptoms of schizophrenia: A multivariate Bayesian meta-analysis
Williams, D. R., & Bürkner, P. (2016) · Psychoneuroendocrinology · doi:10.1016/j.psyneuen.2016.10.013 · Cited by 58
Contemporary statistical inference for infectious disease models using Stan
Chatzilena, A., van Leeuwen, E., Ratmann, O., Baguelin, M., & Demiris, N. (2019) · Epidemics · doi:10.1016/j.epidem.2019.100367 · Cited by 59
“The relatively new software package Stan (Stan Development Team, 2018; Carpenter et al., 2017) provides a generic interface to implementing both HMC and VB, freeing end-users from the challenge of implementing their own computational HMC and VB routines.”
MethodImproving estimates of the state of global fisheries depends on better data
Ovando, D., Hilborn, R. W., Monnahan, C. C., Rudd, M. B., Sharma, R., Thorson, J. T., Rousseau, Y., & Ye, Y. (2021) · Fish and Fisheries · doi:10.1111/faf.12593 · Cited by 58
Wilde, H., Mellan, T. A., Hawryluk, I., Dennis, J., Denaxas, S., Pagel, C., Duncan, A., Bhatt, S., Flaxman, S. R., Mateen, B. A., et al. (2021) · BMC Medicine · doi:10.1186/s12916-021-02096-0 · Cited by 55
Application of three approaches for quantitative AOP development to renal toxicity
Zgheib, E., Gao, W., Limonciel, A., et al. (2019) · Computational Toxicology · doi:10.1016/j.comtox.2019.02.001 · Cited by 57
“For the BN qAOP, posterior parameter distributions were obtained by Hamiltonian MCMC, using the Stan software [48].”
MethodTesting asteroseismology with Gaia DR2: hierarchical models of the Red Clump
Hall, O. J., Davies, G. R., Elsworth, Y., Miglio, A., Bedding, T. R., Brown, A. G. A., Khan, S., Hawkins, K., García, R. A., Chaplin, W. J., et al. (2019) · Monthly Notices of the Royal Astronomical Society · doi:10.1093/mnras/stz1092 · Cited by 57
“Software: This work has made use of the following software not cited in the text: astropy (Astropy Collaboration et al. 2013, 2018), corner (Foreman-Mackey 2016), daft (Foreman-Mackey & Hogg 2012), PyStan (Stan Development Team 2018), Stan (Carpenter et al. 2017), scikitlearn (Pedregosa et al. 2011), iPython (Pérez &…”
BackgroundInfluentialManlove, K. R., Branan, M., Baker, K., et al. (2019) · Preventive Veterinary Medicine · doi:10.1016/j.prevetmed.2019.04.006 · Cited by 57
“The bayes_glm function fits Bayesian generalized linear regression models in the Stan programming language (Carpenter et al., 2017), which uses adaptive Hamiltonian Monte Carlo sampling to simulate draws from the posterior distribution.”
MethodDo voices carry valid information about a speaker’s personality?
Stern, J., Schild, C., Jones, B. C., DeBruine, L. M., Hahn, A. C., Puts, D. A., Zettler, I., Kordsmeyer, T. L., Feinberg, D. R., Zamfir, D., et al. (2021) · Journal of Research in Personality · doi:10.1016/j.jrp.2021.104092 · Cited by 59
“Stan: A probabilistic programming language.”
MethodInfluentialQuantifying a Novel Climate Through Changes in PDO‐Climate and PDO‐Salmon Relationships
Litzow, M. A., Malick, M. J., Bond, N. A., Cunningham, C. J., Gosselin, J. L., & Ward, E. J. (2020) · Geophysical Research Letters · doi:10.1029/2020gl087972 · Cited by 60
Reukers, D. F. M., van Boven, M., Meijer, A., et al. (2021) · Clinical Infectious Diseases · doi:10.1093/cid/ciab237 · Cited by 57
A confirmation bias in perceptual decision-making due to hierarchical approximate inference
Lange, R. D., Chattoraj, A., Beck, J. M., Yates, J. L., & Haefner, R. M. (2021) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1009517 · Cited by 67
Hotspots of biogeochemical activity linked to aridity and plant traits across global drylands
Eldridge, D. J., Ding, J., Dorrough, J. W., Delgado‐Baquerizo, M., Sala, O. E., Gross, N., Bagousse‐Pinguet, Y. L., Mallen‐Cooper, M., Sáiz, H., Asensio, S., et al. (2024) · Nature Plants · doi:10.1038/s41477-024-01670-7 · Cited by 69
“Standardized dung mass (dung mass in a plot/maximum dung mass within the site) was used as a measure of recent grazing pressure.”
MethodInfluentialApproximate Computing: Concepts, Architectures, Challenges, Applications, and Future Directions
Dalloo, A. M., Humaidi, A. J., Mhdawi, A. K. A., & Al-Raweshidy, H. S. (2024) · IEEE Access · doi:10.1109/access.2024.3467375 · Cited by 57
Beninde, J., Feldmeier, S., Veith, M., & Hochkirch, A. (2018) · Proceedings of the Royal Society B Biological Sciences · doi:10.1098/rspb.2018.0143 · Cited by 59
Metadata
- 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
- 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)
- 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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