Bayesian Analysis

Objective Bayesian Analysis for Gaussian Hierarchical Models with Intrinsic Conditional Autoregressive Priors

Matthew J. Keefe, Marco A. R. Ferreira, and Christopher T. Franck

Full-text: Open access


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.

Article information

Bayesian Anal., Volume 14, Number 1 (2019), 181-209.

First available in Project Euclid: 19 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H11: Directional data; spatial statistics 62F15: Bayesian inference
Secondary: 62M30: Spatial processes

conditional autoregressive hierarchical models objective prior reference prior spatial statistics

Creative Commons Attribution 4.0 International License.


Keefe, Matthew J.; Ferreira, Marco A. R.; Franck, Christopher T. Objective Bayesian Analysis for Gaussian Hierarchical Models with Intrinsic Conditional Autoregressive Priors. Bayesian Anal. 14 (2019), no. 1, 181--209. doi:10.1214/18-BA1107.

Export citation


  • Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2014). Hierarchical Modeling and Analysis for Spatial Data. CRC Press.
  • 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.
  • Berger, J. (2006). “The case for objective Bayesian analysis.” Bayesian Analysis, 1(3): 385–402.
  • 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.
  • 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.
  • Bernardo, J. and Smith, A. (1994). Bayesian Theory. New York: Wiley.
  • Besag, J. (1974). “Spatial interaction and the statistical analysis of lattice systems.” Journal of the Royal Statistical Society. Series B (Methodological), 192–236.
  • Besag, J. and Kooperberg, C. (1995). “On conditional and intrinsic autoregressions.” Biometrika, 82(4): 733–746.
  • 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.
  • 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.
  • 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.
  • 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.
  • Bureau of Labor Statistics (2012). “Local Area Unemployment Statistics.” Accessed: 2014-07-14.
  • Clayton, D. and Kaldor, J. (1987). “Empirical Bayes estimates of age-standardized relative risks for use in disease mapping.” Biometrics, 43(3): 671–681.
  • De Oliveira, V. (2007). “Objective Bayesian analysis of spatial data with measurement error.” Canadian Journal of Statistics, 35(2): 283–301.
  • De Oliveira, V. (2012). “Bayesian analysis of conditional autoregressive models.” Annals of the Institute of Statistical Mathematics, 64(1): 107–133.
  • 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.
  • Dietrich, C. (1991). “Modality of the restricted likelihood for spatial Gaussian random fields.” Biometrika, 78(4): 833–839.
  • Ferreira, M. A. R. (2018). “The Limiting Distribution of the Gibbs Sampler for the Intrinsic Conditional Autoregressive Model.” Technical report, Department of Statistics, Virginia Tech.
  • 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.
  • 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.
  • 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.
  • 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.
  • Firth, D. (1993). “Bias reduction of maximum likelihood estimates.” Biometrika, 80: 27–38.
  • 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.
  • Gamerman, D. and Lopes, H. F. (2006). Markov chain Monte Carlo: Stochastic Simulation for Bayesian Inference. CRC Press.
  • 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.
  • Gelman, A. and Rubin, D. B. (1992). “Inference from iterative simulation using multiple sequences.” Statistical Science, 7(4): 457–472.
  • Gilks, W. R., Best, N., and Tan, K. (1995). “Adaptive rejection Metropolis sampling within Gibbs sampling.” Applied Statistics, 44(4): 455–472.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Kuo, B.-S. (1999). “Asymptotics of ML estimator for regression models with a stochastic trend component.” Econometric Theory, 15(01): 24–49.
  • Lavine, M. L. and Hodges, J. S. (2012). “On rigorous specification of ICAR models.” The American Statistician, 66(1): 42–49.
  • Lee, D. (2013). “CARBayes: An R package for Bayesian spatial modeling with conditional autoregressive priors.” Journal of Statistical Software, 55(13): 1–24.
  • 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.
  • 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.
  • 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.
  • Muirhead, R. J. (2009). Aspects of Multivariate Statistical Theory, volume 197. John Wiley & Sons.
  • 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.
  • Penrose, R. (1955). “A generalized inverse for matrices.” In Mathematical Proceedings of the Cambridge Philosophical Society, volume 51, 406–413. Cambridge Univ Press.
  • 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.
  • Ren, C. and Sun, D. (2013). “Objective Bayesian analysis for CAR models.” Annals of the Institute of Statistical Mathematics, 65(3): 457–472.
  • Ren, C. and Sun, D. (2014). “Objective Bayesian analysis for autoregressive models with nugget effects.” Journal of Multivariate Analysis, 124: 260–280.
  • Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. Springer Science & Business Media, 2nd edition.
  • 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.
  • Sun, D., Tsutakawa, R. K., and Speckman, P. L. (1999). “Posterior distribution of hierarchical models using CAR (1) distributions.” Biometrika, 86(2): 341–350.
  • Verbyla, A. P. (1990). “A conditional derivation of residual maximum likelihood.” Australian Journal of Statistics, 32(2): 227–230.

Supplemental materials