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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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MethodSelf-citationHierarchical generalized additive models in ecology: an introduction with mgcv
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“Similarly, the brms package ( Bürkner, 2017 ), which can fi t complex statistical models using the Bayesian software Stan ( Carpenter et al., 2017 ) allows for the inclusion of mgcv -style smooth terms as part of the model speci fi cation.”
MethodBayesian statistics and modelling
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Self-citationThe JASP guidelines for conducting and reporting a Bayesian analysis
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A global synthesis reveals biodiversity-mediated benefits for crop production
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Ultra-large library docking for discovering new chemotypes
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“To sample curves from the posterior distribution given the prior distribution and given the results of testing the 549 compounds, we used Hamiltonian Monte Carlo with no-u-turn sampling with Stan 71 (four chains of 50,000 warm-up and 50,000 sampling steps each and adapt_delta = 0.99 and max_treedepth = 12 control para…”
InfluentialIndices of Effect Existence and Significance in the Bayesian Framework
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Dominantly Inherited Alzheimer Network, Preische, O., Schultz, S. A., Apel, A., Kuhle, J., Kaeser, S. A., Barro, C., Gräber, S., Kuder-Buletta, E., LaFougere, C., et al. (2019) · Nature Medicine · doi:10.1038/s41591-018-0304-3 · Cited by 927
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Hands-On Bayesian Neural Networks—A Tutorial for Deep Learning Users
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“The documentation of the probabilistic programming language Stan [49] provides examples on how to choose σ knowing the expected scale of the considered parameters [50].”
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Bayesian Item Response Modeling in R with brms and Stan
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“pt_delta reduces or ideally eliminates the number of \divergent transition" that indicate problems of the sampler exploring the full posterior distribution and thus bias the posterior estimates (Carpenter et al. 2017;Homan and Gelman 2014). From the standard outputs (not shown here), we can see that the model has…”
MethodAnalyzing ordinal data with metric models: What could possibly go wrong?
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Pyro: Deep Universal Probabilistic ProgrammingPreprint
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“Stan (Carpenter et al. (2017)) is a domain-specific language designed for describing a restricted class of probabilistic programs and performing high-quality automated inference in those models. Church (Goodman et al. (2008)), a probabilistic dialect of Scheme, was an early universal probabilistic programming language…”
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“Although a number of software packages in the R statistical programming environment (R Core Team, 2017) allow modeling ordinal responses, here we use the brms (Bayesian regression models using ‘Stan’) package (Bürkner, 2017, 2018; Carpenter et al., 2017), for two main reasons.”
MethodEstimating infectiousness throughout SARS-CoV-2 infection course
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“Popular probabilistic programming tools include but are not limited to: Stan [28], which covers a large range of advanced VI and MCMC methods, Infer.”
MethodArviZ a unified library for exploratory analysis of Bayesian models in Python
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Power analysis of single-cell RNA-sequencing experiments
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Measurements of the Hubble Constant: Tensions in Perspective
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DeepPoseKit, a software toolkit for fast and robust animal pose estimation using deep learning
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Ecosystem decay exacerbates biodiversity loss with habitat loss
Chase, J. M., Blowes, S. A., Knight, T. M., Gerstner, K., & May, F. (2020) · Nature · doi:10.1038/s41586-020-2531-2 · Cited by 519
Multidimensional tropical forest recovery
Poorter, L., Craven, D., Jakovac, C. C., van der Sande, M. T., Amissah, L., Bongers, F., Chazdon, R. L., Farrior, C. E., Kambach, S., Meave, J. A., et al. (2021) · Science · doi:10.1126/science.abh3629 · Cited by 550
Souporcell: robust clustering of single-cell RNA-seq data by genotype without reference genotypes
Heaton, H., Talman, A. M., Knights, A., Imaz, M., Gaffney, D. J., Durbin, R., Hemberg, M., & Lawniczak, M. K. N. (2020) · Nature Methods · doi:10.1038/s41592-020-0820-1 · Cited by 506
“This model is implemented in the domain-specific language for probabilistic models, STAN 16 , and it solves for the maximum-likelihood soup fraction with gradient descent.”
MethodRelativistic redshift of the star S0-2 orbiting the Galactic Center supermassive black hole
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Health-Status Outcomes with Invasive or Conservative Care in Coronary Disease
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The world’s user-generated road map is more than 80% complete
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“The model is estimated in a Bayesian framework using the open-source PyStan software [46].”
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The world’s growing municipal solid waste: trends and impacts
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“All regressions were undertaken through the brms package (Bürkner 2017) as an interface to the Bayesian inference engine Stan (Carpenter et al 2017).”
MethodFirst M87 Event Horizon Telescope Results. VII. Polarization of the Ring
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COVID-19 Government Response Event Dataset (CoronaNet v.1.0)
Cheng, C. I.-F., Barceló, J., Hartnett, A. S., Kubinec, R., & Messerschmidt, L. (2020) · Nature Human Behaviour · doi:10.1038/s41562-020-0909-7 · Cited by 465
Global shifts in the phenological synchrony of species interactions over recent decades
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“Given that none of the synchrony change estimates could be less than zero (given that we worked with the absolute value of synchrony change), we used a truncated normal distribution for the overall (μα) and species-level (αs) intercept parameters defined in the Stan environment (SI Appendix, section 5, model 4).”
MethodInfluentialSelf-citationSpatiotemporal dynamics of molecular pathology in amyotrophic lateral sclerosis
Maniatis, S., Äijö, T., Vickovic, S., Braine, C., Kang, K., Mollbrink, A., Fagegaltier, D., Andrusivová, Ž., Saarenpää, S., Saiz-Castro, G., et al. (2019) · Science · doi:10.1126/science.aav9776 · Cited by 429
Extracting distribution parameters from multiple uncertain observations with selection biases
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“Code and results implementing this model in the stan sampler (Carpenter et al. 2017) can be found at https://github.com/farr/SelectionExample.”
BackgroundCanopy mortality has doubled in Europe’s temperate forests over the last three decades
Senf, C., Dirk Pflugmacher, A., Yang, Z., Sebald, J., Knorn, J., Neumann, M., Hostert, P., & Seidl, R. (2018) · Nature Communications · doi:10.1038/s41467-018-07539-6 · Cited by 401
Data Integration for Large-Scale Models of Species Distributions
Isaac, N. J. B., Jarzyna, M. A., Keil, P., Dambly, L. I., Boersch‐Supan, P. H., Browning, E., Freeman, S. N., Golding, N., Guillera‐Arroita, G., Henrys, P. A., et al. (2019) · Trends in Ecology & Evolution · doi:10.1016/j.tree.2019.08.006 · Cited by 411
“Newer MCMC software, such as Stan [64] and greta [65], are potentially much quicker, but few comparisons for large ecological datasets exist.”
MethodA systematic review of Bayesian articles in psychology: The last 25 years.
van de Schoot, R., Winter, S. D., Ryan, O., Zondervan-Zwijnenburg, M. A. J., & Depaoli, S. A. (2017) · Psychological Methods · doi:10.1037/met0000100 · Cited by 401
Uncovering novel mutational signatures by de novo extraction with SigProfilerExtractor
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“SigFit uses a previously developed nonnegative factorization method.(29) In contrast, EMu, SA, SigneR, and SigProfilerExtractor provide original implementations of their factorization algorithms (Table 1).”
Ahn, W., Haines, N., & Zhang, L. (2017) · Computational Psychiatry · doi:10.1162/cpsy_a_00002 · Cited by 385
“Then we illustrate how we programmed our models using the Stan software package (Carpenter et al., 2016) and how we formulated hierarchical structures for various types of model parameters (see Performing Hierarchial Bayesian Analysis With Stan).”
MethodInfluentialEvent Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., Alef, W., Algaba, J. C., Anantua, R., Asada, K., Azulay, R., Bach, U., Baczko, A.-K., et al. (2022) · The Astrophysical Journal Letters · doi:10.3847/2041-8213/ac6736 · Cited by 369
“…sampling scheme to explore the posterior space, employing a parallel tempering scheme (Syed et al. 2022) to ensure traversal over the entire prior volume and the Hamiltonian Monte Carlo sampling kernel from the Stan package (Carpenter et al. 2017) to efficiently sample within each tempering level.”
MethodGenome-wide gene expression tuning reveals diverse vulnerabilities of M. tuberculosis
Bosch, B., DeJesus, M. A., Poulton, N. C., Zhang, W., Engelhart, C. A., Zaveri, A., Lavalette, S., Ruecker, N., Trujillo, C., Wallach, J. B., et al. (2021) · Cell · doi:10.1016/j.cell.2021.06.033 · Cited by 316
blavaan : Bayesian Structural Equation Models via Parameter Expansion
Merkle, E. C., & Rosseel, Y. (2018) · Journal of Statistical Software · doi:10.18637/jss.v085.i04 · Cited by 360
A review of automatic differentiation and its efficient implementation
Margossian, C. C. (2019) · Wiley Interdisciplinary Reviews Data Mining and Knowledge Discovery · doi:10.1002/widm.1305 · Cited by 344
“NUTS, supported by AD, indeed provides the algorithmic foundation for the probabilistic languages Stan (Carpenter et al., 2017) and PyMC3 (Salvatier, Wiecki, & Fonnesbeck, 2016).”
BackgroundViral Dynamics of SARS-CoV-2 Variants in Vaccinated and Unvaccinated Persons
Kissler, S. M., Fauver, J. R., Mack, C., Tai, C. G., Breban, M. I., Watkins, A. E., Samant, R. M., Anderson, D. J., Metti, J., Khullar, G., et al. (2021) · New England Journal of Medicine · doi:10.1056/nejmc2102507 · Cited by 335
Age groups that sustain resurging COVID-19 epidemics in the United States
Monod, M., Blenkinsop, A., Xi, X., Hebert, D. J., Bershan, S., Tietze, S., Baguelin, M., Bradley, V. C., Chen, Y., Coupland, H., et al. (2021) · Science · doi:10.1126/science.abe8372 · Cited by 335
Flynn, D. F. B., & Wolkovich, E. M. (2018) · New Phytologist · doi:10.1111/nph.15232 · Cited by 328
“STAN model code and output provided upon request.”
BackgroundMethodInfluentialPlank, C., Rüther, T., Jahn, L., et al. (2023) · Journal of Power Sources · doi:10.1016/j.jpowsour.2023.233845 · Cited by 327
Time for a change: a tutorial for comparing multiple classifiers through Bayesian analysisPreprint
Benavoli, A., Giorgio Corani, A., Demšar, J., & Marco Zaffalon, A. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1606.04316 · Cited by 324
“org) (Carpenter et al., 2016), a language for Bayesian inference.”
MethodBayesian Brains without Probabilities
Sanborn, A. N., & Chater, N. (2016) · Trends in Cognitive Sciences · doi:10.1016/j.tics.2016.10.003 · Cited by 307
Wilson, C., Strickland, M. S., Hutchings, J., Bianchi, T. S., & Flory, S. L. (2018) · Global Change Biology · doi:10.1111/gcb.14070 · Cited by 331
Inferring Continuous and Discrete Population Genetic Structure Across Space
Bradburd, G. S., Coop, G., & Ralph, P. L. (2018) · Genetics · doi:10.1534/genetics.118.301333 · Cited by 310
A psychometrics of individual differences in experimental tasks
Rouder, J. N., & Haaf, J. M. (2019) · Psychonomic Bulletin & Review · doi:10.3758/s13423-018-1558-y · Cited by 321
A genetic, genomic, and computational resource for exploring neural circuit function
Davis, F. P., Nern, A., Picard, S., Reiser, M. B., Rubin, G. M., Eddy, S. R., & Henry, G. L. (2020) · eLife · doi:10.7554/elife.50901 · Cited by 268
Shih, D. J. H., Nayyar, N., Bihun, I. V., Dagogo‐Jack, I., Gill, C. M., Aquilanti, E., Bertalan, M. S., Kaplan, A. B., D’Andrea, M. R., Chukwueke, U. N., et al. (2020) · Nature Genetics · doi:10.1038/s41588-020-0592-7 · Cited by 296
Blair, G., Coppock, A., & Moor, M. (2020) · American Political Science Review · doi:10.1017/s0003055420000374 · Cited by 299
“We conduct Bayesian estimation using Stan (Carpenter et al. 2017), adopting the default improper uniform priors for δ , μ , and τ and restricting τ to be non-negative.”
MethodLife histories of myeloproliferative neoplasms inferred from phylogenies
Williams, N., Lee, J., Mitchell, E., Moore, L., Baxter, E. J., Hewinson, J., Dawson, K. J., Menzies, A., Godfrey, A. L., Green, A. R., et al. (2022) · Nature · doi:10.1038/s41586-021-04312-6 · Cited by 285
Stingray: A Modern Python Library for Spectral Timing
Huppenkothen, D., Bachetti, M., Stevens, A., Migliari, S., Balm, P. W., Hammad, O., Khan, U. M., Mishra, H., Rashid, H., Sharma, S., et al. (2019) · The Astrophysical Journal · doi:10.3847/1538-4357/ab258d · Cited by 282
“Constructing a universal modeling framework is a highly non-trivial task, and excellent packages exist for generalpurpose model building (e.g., STAN, Carpenter et al. 2017).”
MethodNguyen, M., La, V., Le, T.-T., & Vuong, Q. (2022) · MethodsX · doi:10.1016/j.mex.2022.101808 · Cited by 269
Population-wide analysis of differences in disease progression patterns in men and women
Westergaard, D., Moseley, P. L., Sørup, F. K. H., Baldi, P., & Brunak, S. (2019) · Nature Communications · doi:10.1038/s41467-019-08475-9 · Cited by 285
Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo
Monnahan, C. C., Thorson, J. T., & Branch, T. A. (2016) · Methods in Ecology and Evolution · doi:10.1111/2041-210x.12681 · Cited by 287
Excessive Neutrophils and Neutrophil Extracellular Traps in COVID-19
Wang, J., Li, Q., Yin, Y., Zhang, Y., Cao, Y., Lin, X., Huang, L., Hoffmann, D. E., Lu, M., & Qiu, Y. (2020) · Frontiers in Immunology · doi:10.3389/fimmu.2020.02063 · Cited by 280
A Tutorial on Conducting and Interpreting a Bayesian ANOVA in JASP
van den Bergh, D., van Doorn, J. B., Marsman, M., Draws, T., van Kesteren, E., Derks, K., Dablander, F., Gronau, Q. F., Kucharský, Š., Gupta, A. R. K. N., et al. (2020) · L’Année psychologique · doi:10.3917/anpsy1.201.0073 · Cited by 267
TensorFlow DistributionsPreprint
Dillon, J. V., Langmore, I., Tran, D., Brevdo, E., Vasudevan, S., Moore, D., Patton, B., Alemi, A., Hoffman, M., & Saurous, R. A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1711.10604 · Cited by 634
“Stan Math [10] is a C++ templated library for numerical and statistical functions, and with automatic differentiation as the backend for the Stan probabilistic programming language [9].”
BackgroundSelf-citationTimeLapse-seq: adding a temporal dimension to RNA sequencing through nucleoside recoding
Schofield, J. A., Duffy, E. E., Kiefer, L. A., Sullivan, M., & Simon, M. D. (2018) · Nature Methods · doi:10.1038/nmeth.4582 · Cited by 254
MERS-CoV spillover at the camel-human interface
Dudas, G., Carvalho, L. M., Rambaut, A., Bedford, T., Somily, A. M., Barry, M., Subaie, S. S. A., BinSaeed, A. A., Alzamil, F. A., Zaher, W., et al. (2018) · eLife · doi:10.7554/elife.31257 · Cited by 269
Union through UNITY: Cosmology with 2000 SNe Using a Unified Bayesian Framework
Rubin, D., Aldering, G., Betoule, M., Fruchter, A., Huang, X., Kim, A., Lidman, C. E., Linder, E. V., Perlmutter, S., Ruiz-Lapuente, P., et al. (2025) · The Astrophysical Journal · doi:10.3847/1538-4357/adc0a5 · Cited by 505
Toward a principled Bayesian workflow in cognitive science.
Schad, D. J., Betancourt, M., & Vasishth, S. (2021) · Psychological Methods · doi:10.1037/met0000275 · Cited by 252
“Much progress has been made in the development of probabilistic programming languages such as Stan (Carpenter et al., 2017), WinBUGS (Lunn, Thomas, Best, & Spiegelhalter, 2000), JASP (JASP Team, 2019), and JAGS (Plummer, 2012).”
BackgroundMethodInfluentialSelf-citationDissecting the null model for biological invasions: A meta-analysis of the propagule pressure effect
Cassey, P., Delean, S., Lockwood, J. L., Sadowski, J. S., & Blackburn, T. M. (2018) · PLoS Biology · doi:10.1371/journal.pbio.2005987 · Cited by 259
“All Bayesian models were fitted using the opensource software Stan [70, 71] and the R package brms [72].”
The ecological and genomic basis of explosive adaptive radiation
McGee, M. D., Borstein, S. R., Meier, J. I., Marques, D. A., Mwaiko, S., Taabu‐Munyaho, A., Kishe, M. A., O’Meara, B. C., Bruggmann, R., Excoffier, L., et al. (2020) · Nature · doi:10.1038/s41586-020-2652-7 · Cited by 254
“Both Bayesian regression analysis 16 and a nonparametric test 17 suggest that the presence of large visually oriented predatory fish 18 and an arid climate are major constraints to speciation rate in cichlids, whereas male-restricted ornamentation (indicative of evolution in response to sexual…”
An interactive framework for whole-brain maps at cellular resolution
Fürth, D., Vaissière, T., Tzortzi, O., Xuan, Y., Märtin, A., Lazaridis, I., Spigolon, G., Fisone, G., Tomer, R., Deisseroth, K., et al. (2018) · Nature Neuroscience · doi:10.1038/s41593-017-0027-7 · Cited by 215
Kissler, S. M., Fauver, J. R., Mack, C. D., Olesen, S. W., Tai, C. G.-L., Shiue, K. Y., Kalinich, C. C., Jednak, S., Ott, I. M., Vogels, C. B. F., et al. (2021) · PLoS Biology · doi:10.1371/journal.pbio.3001333 · Cited by 251
“We estimated the parameters of the regression model by fitting to the available data using a Hamiltonian Monte Carlo algorithm [9] yielding simulated draws from the Bayesian posterior distribution for each parameter.”
MethodDorsal hippocampus contributes to model-based planning
Miller, K. J., Botvinick, M., & Brody, C. D. (2017) · Nature Neuroscience · doi:10.1038/nn.4613 · Cited by 252
Citizen science provides a reliable and scalable tool to track disease-carrying mosquitoes
Palmer, J. R. B., Oltra, A., Collantes, F., Delgado, J. A., Lucientes, J., Delacour‐Estrella, S., Bengoa, M., Eritja, R., & Bartumeus, F. (2017) · Nature Communications · doi:10.1038/s41467-017-00914-9 · Cited by 245
The Generalizability CrisisPreprint
Yarkoni, T. (2019) · PsyArXiv (OSF Preprints) · doi:10.31234/osf.io/jqw35 · Cited by 290
A sensory appendage protein protects malaria vectors from pyrethroids
Ingham, V. A., Anthousi, A., Douris, V., Harding, N. J., Lycett, G. J., Morris, M., VONTAS, J. G., & Ranson, H. (2019) · Nature · doi:10.1038/s41586-019-1864-1 · Cited by 224
Overcoming Equifinality: Leveraging Long Time Series for Stream Metabolism Estimation
Appling, A., Hall, R. O., Yackulic, C. B., & Arroita, M. (2018) · Journal of Geophysical Research Biogeosciences · doi:10.1002/2017jg004140 · Cited by 234
“631 Bayesian models lean heavily on the general-purpose Bayesian modeling software Stan (Carpenter et al., 2017; Stan Development Team, 2016a) via the rstan R interface (Stan Development Team, 2016b).”
Influentialbridgesampling: An R Package for Estimating Normalizing Constants
Gronau, Q. F., Singmann, H., & Wagenmakers, E. (2020) · Journal of Statistical Software · doi:10.18637/jss.v092.i10 · Cited by 323
Evaluating Probabilistic Forecasts with scoringRules
Jordan, A. I., Krüger, F., & Lerch, S. (2019) · Journal of Statistical Software · doi:10.18637/jss.v090.i12 · Cited by 231
“MCMC methods are popular across the disciplines, and many sophisticated R implementations are available (e.g., Kastner 2016; Carpenter et al. 2017).”
Method
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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