The R2D2 Prior for Generalized Linear Mixed Models

References

• Bai, R., and Ghosh, M. (2021), “On the Beta Prime Prior for Scale Parameters in High-Dimensional Bayesian Regression Models,” Statistica Sinica, 31, 1–23. DOI: 10.5705/ss.202019.0037.
   Web of Science ®Google Scholar
• Bhadra, A., Datta, J., Polson, N. G., and Willard, B. (2017), “The Horseshoe + Estimator of Ultra-Sparse Signals,” Bayesian Analysis, 12, 1105–1131. DOI: 10.1214/16-BA1028.
   Web of Science ®Google Scholar
• Bhattacharya, A., Pati, D., Pillai, N. S., and Dunson, D. B. (2015), “Dirichlet–Laplace Priors for Optimal Shrinkage,” Journal of the American Statistical Association, 110, 1479–1490. DOI: 10.1080/01621459.2014.960967.
   PubMed Web of Science ®Google Scholar
• Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A. (2017), “Stan: A Probabilistic Programming Language,” Journal of Statistical Software, 76, 1–32. DOI: 10.18637/jss.v076.i01.
   PubMed Web of Science ®Google Scholar
• Carvalho, C. M., Polson, N. G., and Scott, J. G. (2009), “Handling Sparsity via the Horseshoe,” in Artificial Intelligence and Statistics, pp. 73–80, PMLR.
   Google Scholar
• Carvalho, C. M., Polson, N. G., and Scott, J. G. (2010), “The Horseshoe Estimator for Sparse Signals,” Biometrika, 97, 465–480. DOI: 10.1093/biomet/asq017.
   Web of Science ®Google Scholar
• Coles, S., Bawa, J., Trenner, L., and Dorazio, P. (2001), An Introduction to Statistical Modeling of Extreme Values (Vol. 208), London: Springer.
   Google Scholar
• Cox, D. R., and Snell, E. J. (1989), Analysis of Binary Data, New York: Chapman & Hall.
   Google Scholar
• Diggle, P. J., Tawn, J. A., and Moyeed, R. A. (1998), “Model-based Geostatistics,” Journal of the Royal Statistical Society, Series C, 47, 299–350. DOI: 10.1111/1467-9876.00113.
   Web of Science ®Google Scholar
• Fuglstad, G.-A., Hem, I. G., Knight, A., Rue, H., and Riebler, A. (2020), “Intuitive Joint Priors for Variance Parameters,” Bayesian Analysis, 15, 1109–1137. DOI: 10.1214/19-BA1185.
   Web of Science ®Google Scholar
• Gelman, A. (2006), “Prior Distributions for Variance Parameters in Hierarchical Models,” (comment on article by Browne and Draper), Bayesian Analysis, 1, 515–534. DOI: 10.1214/06-BA117A.
   Web of Science ®Google Scholar
• Gelman, A., Goodrich, B., Gabry, J., and Vehtari, A. (2019), “R-squared for Bayesian Regression Models,” The American Statistician, 73, 307–309. DOI: 10.1080/00031305.2018.1549100.
   Web of Science ®Google Scholar
• Gelman, A., and Hill, J. (2006), Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge: Cambridge University Press.
   Google Scholar
• George, E. I., and McCulloch, R. E. (1993), “Variable Selection via Gibbs Sampling,” Journal of the American Statistical Association, 88, 881–889. DOI: 10.1080/01621459.1993.10476353.
   Web of Science ®Google Scholar
• Hamura, Y., Irie, K., and Sugasawa, S. (2022), “On Global-Local Shrinkage Priors for Count Data,” Bayesian Analysis, 17, 545–564. DOI: 10.1214/21-BA1263.
   Web of Science ®Google Scholar
• Hans, C. (2009), “Bayesian Lasso Regression,” Biometrika, 96, 835–845. DOI: 10.1093/biomet/asp047.
   Web of Science ®Google Scholar
• Hastie, T. J. (2017), Generalized Additive Models, Boca Raton, FL: Routledge.
   Google Scholar
• Hodges, J. S., and Sargent, D. J. (2001), “Counting Degrees of Freedom in Hierarchical and Other Richly-Parameterised Models,” Biometrika, 88, 367–379. DOI: 10.1093/biomet/88.2.367.
   Web of Science ®Google Scholar
• Klein, N., Carlan, M., Kneib, T., Lang, S., and Wagner, H. (2021), “Bayesian Effect Selection in Structured Additive Distributional Regression Models,” Bayesian Analysis, 16, 545–573. DOI: 10.1214/20-BA1214.
   Web of Science ®Google Scholar
• Lewis, C. M., and Knight, J. (2012), “Introduction to Genetic Association Studies,” Cold Spring Harbor Protocols, 2012, pdb–top068163. DOI: 10.1101/pdb.top068163.
   Google Scholar
• Lindgren, F., and Rue, H. (2015), “Bayesian Spatial Modelling with R-INLA,” Journal of Statistical Software, 63, 1–25. DOI: 10.18637/jss.v063.i19.
   Web of Science ®Google Scholar
• Lindley, D. V. (1957), “A Statistical Paradox,” Biometrika, 44, 187–192. DOI: 10.1093/biomet/44.1-2.187.
   Web of Science ®Google Scholar
• McFadden, D. (1973), “Conditional Logit Analysis of Qualitative Choice Behavior,” Frontiers in Econometrics, 105–142.
   Google Scholar
• Morokoff, W. J., and Caflisch, R. E. (1995), “Quasi-Monte Carlo Integration,” Journal of Computational Physics, 122, 218–230. DOI: 10.1006/jcph.1995.1209.
   Web of Science ®Google Scholar
• Park, T., and Casella, G. (2008), “The Bayesian Lasso,” Journal of the American Statistical Association, 103, 681–686. DOI: 10.1198/016214508000000337.
   Web of Science ®Google Scholar
• Perez, M.-E., Pericchi, L., and Ramirez, I. (2017), “The Scaled Beta2 Distribution as a Robust Prior for Scales,” Bayesian Analysis, 12, 615–637. DOI: 10.1214/16-BA1015.
   Web of Science ®Google Scholar
• Pérez-Enciso, M., and Tenenhaus, M. (2003), “Prediction of Clinical Outcome with Microarray Data: A Partial Least Squares Discriminant Analysis (pls-da) Approach,” Human Genetics, 112, 581–592.
   PubMed Web of Science ®Google Scholar
• Perou, C. M., Sørlie, T., Eisen, M. B., Van De Rijn, M., Jeffrey, S. S., Rees, C. A., Pollack, J. R., Ross, D. T., Johnsen, H., Akslen, L. A., et al. (2000), “Molecular Portraits of Human Breast Tumours,” Nature, 406, 747–752. DOI: 10.1038/35021093.
   PubMed Web of Science ®Google Scholar
• Plummer, M., Stukalov, A., Denwood, M., and Plummer, M. M. (2016), Package “rjags”, Vienna, Austria.
   Google Scholar
• Polson, N. G., and Scott, J. G. (2012), “On the Half-Cauchy Prior for a Global Scale Parameter,” Bayesian Analysis, 7, 887–902. DOI: 10.1214/12-BA730.
   Web of Science ®Google Scholar
• Polson, N. G., Scott, J. G., Clarke, B. S., and Severinski, C. (2012), “Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction,” in Bayesian Statistics 9 eds. J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, and M. West, pp. 501–538, Oxford: Oxford University Press.
   Google Scholar
• Rényi, A. (1961), “On Measures of Entropy and Information,” in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pp. 547–561, University of California Press.
   Google Scholar
• Ribeiro Jr, P. J., Diggle, P. J., Ribeiro Jr, M. P. J., and Suggests, M. (2007), “The geoR Package,” R News, 1, 14–18.
   Google Scholar
• Ročková, V., and George, E. I. (2018), “The Spike-and-Slab Lasso,” Journal of the American Statistical Association, 113, 431–444. DOI: 10.1080/01621459.2016.1260469.
   Web of Science ®Google Scholar
• Rohart, F., Gautier, B., Singh, A., and Lê Cao, K.-A. (2017), “mixomics: An r Package for ‘Omics Feature Selection and Multiple Data Integration,” PLoS Computational Biology, 13, e1005752. DOI: 10.1371/journal.pcbi.1005752.
   PubMed Web of Science ®Google Scholar
• Searle, S. R., Casella, G., and McCulloch, C. E. (2009), Variance Components, Hoboken, NJ: Wiley.
   Google Scholar
• Simpson, D., Rue, H., Riebler, A., Martins, T. G., Sørbye, S. H., et al. (2017), “Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors,” Statistical Science, 32, 1–28. DOI: 10.1214/16-STS576.
   Web of Science ®Google Scholar
• Spiegelhalter, D., Thomas, A., Best, N., and Lunn, D. (2003), “Winbugs User Manual.”
   Google Scholar
• Thomson, M. C., Connor, S. J., D’Alessandro, U., Rowlingson, B., Diggle, P., Cresswell, M., and Greenwood, B. (1999), “Predicting Malaria Infection in Gambian Children from Satellite Data and Bed Net Use Surveys: The Importance of Spatial Correlation in the Interpretation of Results,” The American Journal of Tropical Medicine and Hygiene, 61, 2–8. DOI: 10.4269/ajtmh.1999.61.2.
   PubMed Web of Science ®Google Scholar
• Zhang, Y., and Bondell, H. D. (2018), “Variable Selection via Penalized Credible Regions with Dirichlet–Laplace Global-Local Shrinkage Priors,” Bayesian Analysis, 13, 823–844. DOI: 10.1214/17-BA1076.
   Web of Science ®Google Scholar
• Zhang, Y. D., Naughton, B. P., Bondell, H. D., and Reich, B. J. (2022), “Bayesian Regression Using a Prior on the Model Fit: The r2-d2 Shrinkage Prior,” Journal of the American Statistical Association, 117, 862–874. DOI: 10.1080/01621459.2020.1825449.
   Web of Science ®Google Scholar