Objective Bayesian analysis for Gaussian hierarchical models with intrinsic conditional autoregressive priors.

*(English)*Zbl 1409.62187Summary: Bayesian hierarchical models are commonly used for modeling spatially correlated areal data. However, choosing appropriate prior distributions for the parameters in these models is necessary and sometimes challenging. In particular, an intrinsic conditional autoregressive (CAR) hierarchical component is often used to account for spatial association. Vague proper prior distributions have frequently been used for this type of model, but this requires the careful selection of suitable hyperparameters. In this paper, we derive several objective priors for the Gaussian hierarchical model with an intrinsic CAR component and discuss their properties. We show that the independence Jeffreys and Jeffreys-rule priors result in improper posterior distributions, while the reference prior results in a proper posterior distribution. We present results from a simulation study that compares frequentist properties of Bayesian procedures that use several competing priors, including the derived reference prior. We demonstrate that using the reference prior results in favorable coverage, interval length, and mean squared error. Finally, we illustrate our methodology with an application to 2012 housing foreclosure rates in the 88 counties of Ohio.

##### MSC:

62M30 | Inference from spatial processes |

62F15 | Bayesian inference |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62P20 | Applications of statistics to economics |

##### Keywords:

conditional autoregressive model; hierarchical models; objective prior; reference prior; spatial statistics##### References:

[1] | Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2014). Hierarchical Modeling and Analysis for Spatial Data. CRC Press. · Zbl 1358.62009 |

[2] | Bell, B. S. and Broemeling, L. D. (2000). “A Bayesian analysis for spatial processes with application to disease mapping.” Statistics in Medicine, 19(7): 957–974. |

[3] | Berger, J. (2006). “The case for objective Bayesian analysis.” Bayesian Analysis, 1(3): 385–402. · Zbl 1331.62042 |

[4] | Berger, J. O., De Oliveira, V., and Sansó, B. (2001). “Objective Bayesian analysis of spatially correlated data.” Journal of the American Statistical Association, 96(456): 1361–1374. · Zbl 1051.62095 |

[5] | Bernardinelli, L., Clayton, D., and Montomoli, C. (1995). “Bayesian estimates of disease maps: how important are priors?” Statistics in Medicine, 14(21–22): 2411–2431. |

[6] | Bernardo, J. and Smith, A. (1994). Bayesian Theory. New York: Wiley. · Zbl 0796.62002 |

[7] | Besag, J. (1974). “Spatial interaction and the statistical analysis of lattice systems.” Journal of the Royal Statistical Society. Series B (Methodological), 192–236. · Zbl 0327.60067 |

[8] | Besag, J. and Kooperberg, C. (1995). “On conditional and intrinsic autoregressions.” Biometrika, 82(4): 733–746. · Zbl 0899.62123 |

[9] | Besag, J., York, J., and Mollié, A. (1991). “Bayesian image restoration, with two applications in spatial statistics.” Annals of the Institute of Statistical Mathematics, 43(1): 1–20. · Zbl 0760.62029 |

[10] | Best, N., Richardson, S., and Thomson, A. (2005). “A comparison of Bayesian spatial models for disease mapping.” Statistical Methods in Medical Research, 14(1): 35–59. · Zbl 1057.62097 |

[11] | Best, N., Waller, L., Thomas, A., Conlon, E., and Arnold, R. (1999). “Bayesian models for spatially correlated disease and exposure data.” In Bayesian Statistics 6: Proceedings of the Sixth Valencia International Meeting, volume 6, 131. Oxford University Press. · Zbl 0973.62020 |

[12] | Bouckaert, R., Heled, J., Kühnert, D., Vaughan, T., Wu, C.-H., Xie, D., Suchard, M., Rambaut, A., and Drummond, A. J. (2014). “BEAST 2: A Software Platform for Bayesian Evolutionary Analysis.” PLoS Computational Biology, 10(4): e1003537. |

[13] | Bureau of Labor Statistics (2012). “Local Area Unemployment Statistics.” http://www.bls.gov/lau/. Accessed: 2014-07-14. |

[14] | Clayton, D. and Kaldor, J. (1987). “Empirical Bayes estimates of age-standardized relative risks for use in disease mapping.” Biometrics, 43(3): 671–681. |

[15] | De Oliveira, V. (2007). “Objective Bayesian analysis of spatial data with measurement error.” Canadian Journal of Statistics, 35(2): 283–301. · Zbl 1129.62086 |

[16] | De Oliveira, V. (2012). “Bayesian analysis of conditional autoregressive models.” Annals of the Institute of Statistical Mathematics, 64(1): 107–133. · Zbl 1238.62028 |

[17] | De Oliveira, V. and Ferreira, M. A. R. (2011). “Maximum likelihood and restricted maximum likelihood estimation for a class of Gaussian Markov random fields.” Metrika, 74(2): 167–183. · Zbl 1230.62106 |

[18] | Dietrich, C. (1991). “Modality of the restricted likelihood for spatial Gaussian random fields.” Biometrika, 78(4): 833–839. · Zbl 0850.62699 |

[19] | Ferreira, M. A. R. and De Oliveira, V. (2007). “Bayesian reference analysis for Gaussian Markov random fields.” Journal of Multivariate Analysis, 98(4): 789–812. · Zbl 1118.62100 |

[20] | Ferreira, M. A. R., Holan, S. H., and Bertolde, A. I. (2011). “Dynamic multiscale spatiotemporal models for Gaussian areal data.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(5): 663–688. · Zbl 1228.62116 |

[21] | Ferreira, M. A. R. and Salazar, E. (2014). “Bayesian reference analysis for exponential power regression models.” Journal of Statistical Distributions and Applications, 1(1): 1–12. · Zbl 1327.62142 |

[22] | Ferreira, M. A. R. and Suchard, M. A. (2008). “Bayesian Analysis of Elapsed Times in Continuous-Time Markov Chains.” Canadian Journal of Statistics, 36: 355–368. · Zbl 1153.62019 |

[23] | Firth, D. (1993). “Bias reduction of maximum likelihood estimates.” Biometrika, 80: 27–38. · Zbl 0769.62021 |

[24] | Fonseca, T. C. O., Ferreira, M. A. R., and Migon, H. S. (2008). “Objective Bayesian analysis for the Student-\(t\) regression model.” Biometrika, 95(2): 325–333. · Zbl 1400.62260 |

[25] | Gamerman, D. and Lopes, H. F. (2006). Markov chain Monte Carlo: Stochastic Simulation for Bayesian Inference. CRC Press. · Zbl 1137.62011 |

[26] | Gelfand, A. E. and Smith, A. F. (1990). “Sampling-based approaches to calculating marginal densities.” Journal of the American Statistical Association, 85(410): 398–409. · Zbl 0702.62020 |

[27] | Gelman, A. and Rubin, D. B. (1992). “Inference from iterative simulation using multiple sequences.” Statistical Science, 7(4): 457–472. · Zbl 1386.65060 |

[28] | Gilks, W. R., Best, N., and Tan, K. (1995). “Adaptive rejection Metropolis sampling within Gibbs sampling.” Applied Statistics, 44(4): 455–472. · Zbl 0893.62110 |

[29] | Goicoa, T., Ugarte, M., Etxeberria, J., and Militino, A. (2016). “Age–space–time CAR models in Bayesian disease mapping.” Statistics in Medicine, 35(14): 2391–2405. |

[30] | Hodges, J. S., Carlin, B. P., and Fan, Q. (2003). “On the precision of the conditionally autoregressive prior in spatial models.” Biometrics, 59(2): 317–322. · Zbl 1210.62128 |

[31] | Keefe, M. J., Ferreira, M. A. R., and Franck, C. T. (2018). “On the formal specification of sum-zero constrained intrinsic conditional autoregressive models.” Spatial Statistics, 24: 54–65. |

[32] | Keefe, M. J., Ferreira, M. A. R., and Franck, C. T. (2019). “Supplementary Material of Objective Bayesian Analysis for Gaussian Hierarchical Models with Intrinsic Conditional Autoregressive Priors.” Bayesian Analysis. · Zbl 1409.62187 |

[33] | Keefe, M. J., Franck, C. T., and Woodall, W. H. (2017). “Monitoring foreclosure rates with a spatially risk-adjusted Bernoulli CUSUM chart for concurrent observations.” Journal of Applied Statistics, 44(2): 325–341. |

[34] | Kuo, B.-S. (1999). “Asymptotics of ML estimator for regression models with a stochastic trend component.” Econometric Theory, 15(01): 24–49. · Zbl 0962.62119 |

[35] | Lavine, M. L. and Hodges, J. S. (2012). “On rigorous specification of ICAR models.” The American Statistician, 66(1): 42–49. |

[36] | Lee, D. (2013). “CARBayes: An R package for Bayesian spatial modeling with conditional autoregressive priors.” Journal of Statistical Software, 55(13): 1–24. |

[37] | Liu, Z., Berrocal, V. J., Bartsch, A. J., and Johnson, T. D. (2016). “Pre-surgical fMRI Data Analysis Using a Spatially Adaptive Conditionally Autoregressive Model.” Bayesian Analysis, 11: 599–625. · Zbl 1359.62407 |

[38] | Mercer, L. D., Wakefield, J., Pantazis, A., Lutambi, A. M., Masanja, H., and Clark, S. (2015). “Space–time smoothing of complex survey data: Small area estimation for child mortality.” The Annals of Applied Statistics, 9(4): 1889–1905. · Zbl 1397.62461 |

[39] | Moraga, P. and Lawson, A. B. (2012). “Gaussian component mixtures and CAR models in Bayesian disease mapping.” Computational Statistics & Data Analysis, 56(6): 1417–1433. · Zbl 1243.62037 |

[40] | Muirhead, R. J. (2009). Aspects of Multivariate Statistical Theory, volume 197. John Wiley & Sons. · Zbl 0556.62028 |

[41] | Natarajan, R. and McCulloch, C. E. (1998). “Gibbs sampling with diffuse proper priors: A valid approach to data-driven inference?” Journal of Computational and Graphical Statistics, 7(3): 267–277. |

[42] | Penrose, R. (1955). “A generalized inverse for matrices.” In Mathematical Proceedings of the Cambridge Philosophical Society, volume 51, 406–413. Cambridge Univ Press. · Zbl 0065.24603 |

[43] | Reich, B. J., Hodges, J. S., and Zadnik, V. (2006). “Effects of residual smoothing on the posterior of the fixed effects in disease-mapping models.” Biometrics, 62(4): 1197–1206. · Zbl 1114.62124 |

[44] | Ren, C. and Sun, D. (2013). “Objective Bayesian analysis for CAR models.” Annals of the Institute of Statistical Mathematics, 65(3): 457–472. · Zbl 1396.62044 |

[45] | Ren, C. and Sun, D. (2014). “Objective Bayesian analysis for autoregressive models with nugget effects.” Journal of Multivariate Analysis, 124: 260–280. · Zbl 1359.62382 |

[46] | Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. Springer Science & Business Media, 2nd edition. · Zbl 1096.62003 |

[47] | Salazar, E., Ferreira, M. A. R., and Migon, H. S. (2012). “Objective Bayesian analysis for exponential power regression models.” Sankhya – Series B, 74: 107–125. · Zbl 1257.62029 |

[48] | Sun, D., Tsutakawa, R. K., and Speckman, P. L. (1999). “Posterior distribution of hierarchical models using CAR (1) distributions.” Biometrika, 86(2): 341–350. · Zbl 0931.62081 |

[49] | Verbyla, A. P. (1990). “A conditional derivation of residual maximum likelihood.” Australian Journal of Statistics, 32(2): 227–230. |

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