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Posterior propriety of an objective prior for generalized hierarchical normal linear models

Cong Lin ,

School of Statistics, East China Normal University, Shanghai, People's Republic of China

Dongchu Sun ,

School of Statistics, East China Normal University, Shanghai, People's Republic of China;b Department of Statistics, University of Nebraska-Lincoln, Lincoln, NE, USA

Chengyuan Song

Boehringer Ingelheim (China), Shanghai, People's Republic of China

Pages | Received 23 Jan. 2021, Accepted 31 Aug. 2021, Published online: 30 Jul. 2022,
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Bayesian Hierarchical models has been widely used in modern statistical application. To deal with the data having complex structures, we propose a generalized hierarchical normal linear (GHNL) model which accommodates arbitrarily many levels, usual design matrices and ‘vanilla’ covariance matrices. Objective hyperpriors can be employed for the GHNL model to express ignorance or match frequentist properties, yet the common objective Bayesian approaches are infeasible or fraught with danger in hierarchical modelling. To tackle this issue, [Berger, J., Sun, D., & Song, C. (2020b). An objective prior for hyperparameters in normal hierarchical models. Journal of Multivariate Analysis, 178, 104606.] proposed a particular objective prior and investigated its properties comprehensively. Posterior propriety is important for the choice of priors to guarantee the convergence of MCMC samplers. James Berger conjectured that the resulting posterior is proper for a hierarchical normal model with arbitrarily many levels, a rigorous proof of which was not given, however. In this paper, we complete this story and provide an user-friendly guidance. One main contribution of this paper is to propose a new technique for deriving an elaborate upper bound on the integrated likelihood, but also one unified approach to checking the posterior propriety for linear models. An efficient Gibbs sampling method is also introduced and outperforms other sampling approaches considerably.

  • Berger, J. (1980). A robust generalized Bayes estimator and confidence region for a multivariate normal mean. Annals of Statistics8(4), 716–761. 
  • Berger, J., Strawderman, W., & Tang, D. (2005). Posterior propriety and admissibility of hyperpriors in normal hierarchical models. Annals of Statistics33(2), 606–646. 
  • Berger, J., Sun, D., & Song, C. (2020a). Bayesian analysis of the covariance matrix of a multivariate normal distribution with a new class of priors. Annals of Statistics48(4), 2381–2403. 
  • Berger, J., Sun, D., & Song, C. (2020b). An objective prior for hyperparameters in normal hierarchical models. Journal of Multivariate Analysis178(2020), 1–13. 
  • Consonni, G., Fouskakis, D., Liseo, B., & Ntzoufras, I. (2018). Prior distributions for objective Bayesian analysis. Bayesian Analysis13(2), 627–679. 
  • Daniels, M. J., & Kass, R. E. (1999). Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models. Journal of the American Statistical Association.94(448), 1254–1263. 
  • Everson, P. J., & Morris, C. N. (2000). Inference for multivariate normal hierarchical models. Journal of the Royal Statistical Society: Series B62(2), 399–412. 
  • Fourdrinier, D., Strawderman, W. E., & Wells, M. T. (1998). On the construction of Bayes minimax estimators. Annals of Statistics26(2), 660–671. 
  • Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Analysis1(3), 515–534. 
  • Goldstein, H. (2011). Multilevel statistical models (Vol. 922). John Wiley & Sons. 
  • Gustafson, P., Hossain, S., & Macnab, Y. C. (2006). Conservative prior distributions for variance parameters in hierarchical models. Canadian Journal of Statistics34(3), 377–390. 
  • Hobert, J. P., & Casella, G. (1996). The effect of improper priors on Gibbs sampling in hierarchical linear mixed models. Journal of the American Statistical Association91(436), 1461–1473. 
  • Hoff, P. D. (2009b). Simulation of the matrix Bingham-Von Mises-Fisher distribution, with applications to multivariate and relational data. Journal of Computational and Graphical Statistics18(2), 438–456. 
  • Horn, R. A., & Johnson, C. R. (2012). Matrix analysis. Cambridge university press.
  • Lindenberger, U., & Pötter, U. (1998). The complex nature of unique and shared effects in hierarchical linear regression: Implications for developmental psychology. Psychological Methods3(2), 218–230. 
  • Michalak, S. E., & Morris, C. N. (2016). Posterior propriety for hierarchical models with log-Likelihoods that have norm bounds. Bayesian Analysis11(2), 545–571.
  • Raudenbush, S., & Bryk, A. S. (1986). A hierarchical model for studying school effects. Sociology of Education59(1), 1–17. 
  • Shimotsu, K. (2010). Exact local Whittle estimation of fractional integration with unknown mean and time trend. Econometric Theory26(2), 501–540. 
  • Song, C., Sun, D., Fan, K., & Mu, R. (2020). Posterior propriety of an objective prior in a 4-Level normal hierarchical model. Mathematical Problems in Engineering2020 
  • Speckman, P. L., & Sun, D. (2003). Fully Bayesian spline smoothing and intrinsic autoregressive priors. Biometrika90(2), 289–302. 
  • Sun, D., Tsutakawa, R. K., & He, Z. (2001). Propriety of posteriors with improper priors in hierarchical linear mixed models. Statistica Sinica11(1), 77–95. 
  • Xia, A., Ma, H., & Carlin, B. P. (2011). Bayesian hierarchical modeling for detecting safety signals in clinical trials. Journal of Biopharmaceutical Statistics21(5), 1006–1029. 
  • Yang, R., & Berger, J. (1994). Estimation of a covariance matrix using the reference prior. Annals of Statistics22(3), 1195–1211.

To cite this article: Cong Lin, Dongchu Sun & Chengyuan Song (2022): Posterior propriety of an objective prior for generalized hierarchical normal linear models, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2021.1978206

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