Abstract
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.
Highlights
Bayesian hierarchical models with intrinsic conditional autoregressive (CAR) priors are used for many statistical models for spatially dependent data in applications such as disease mapping (Clayton and Kaldor, 1987; Bell and Broemeling, 2000; Moraga and Lawson, 2012; Goicoa et al, 2016), image restoration (Besag et al, 1991), complex survey data (Mercer et al, 2015), and neuroimaging (Liu et al, 2016)
Our reference prior leads to a proper posterior distribution, while the independence Jeffreys and Jeffreys-rule priors lead to improper posterior distributions, unlike De Oliveira (2007)
The CARBayes prior yields wide average interval lengths, potentially detracting from the value of its close-to-nominal coverage. These results demonstrate that, when compared to the two gamma priors, the reference prior leads to favorable coverage and interval length
Summary
Bayesian hierarchical models with intrinsic conditional autoregressive (CAR) priors are used for many statistical models for spatially dependent data in applications such as disease mapping (Clayton and Kaldor, 1987; Bell and Broemeling, 2000; Moraga and Lawson, 2012; Goicoa et al, 2016), image restoration (Besag et al, 1991), complex survey data (Mercer et al, 2015), and neuroimaging (Liu et al, 2016). Bayesian methods have been the dominating paradigm for spatial models including a CAR component (Sun et al, 1999; Hodges et al, 2003; Reich et al, 2006; Banerjee et al, 2014). It is difficult to subjectively choose informative priors for the parameters of the CAR component that are meaningful based on
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