We develop a novel reference prior for Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) random effects. This is particularly important in the context of objective Bayes variable selection with sample size $n$ and $k$ regressors. In this context, a previously published reference prior requires the computation of spectral decompositions of two $n$-dimensional matrices for each model under consideration. As a consequence, for variable selection the computational cost of this previous reference prior grows as $O(n^3 2^k)$. In contrast, our novel reference prior requires the computation of the spectral decomposition of one $n$-dimensional matrix that can be used for all models under consideration. Thus, the computational cost of our novel reference prior grows much slower as $O(n^3)$. Hence, computational savings can be substantial, e.g. in a problem with 10 regressors, when compared to the previously published reference prior, computations based on our novel reference prior are more than 1000 times faster. We provide a proof of the equivalence of the two priors. A simulation study shows that, while both reference priors provide equivalent variable selection results, for large sample sizes computations based on our novel prior are several orders of magnitude faster. Finally, the utility of our novel reference prior is illustrated with a spatial regression study of county-level median household income on socio-economic regressors for 3108 counties in the contiguous United States.

翻译：我们为具有内在条件自回归（ICAR）随机效应的高斯层次模型开发了一种新型参考先验。这在样本量为$n$且包含$k$个回归变量的客观贝叶斯变量选择背景下尤为重要。在此背景下，先前发表的参考先验需要对每个待考虑模型计算两个$n$维矩阵的谱分解。因此，该先前的参考先验在变量选择中的计算成本随$O(n^3 2^k)$增长。相比之下，我们的新型参考先验仅需计算一个$n$维矩阵的谱分解，该矩阵可用于所有待考虑模型。因此，新型参考先验的计算成本增长缓慢得多，为$O(n^3)$。例如，在包含10个回归变量的问题中，与先前发表的参考先验相比，基于新型参考先验的计算速度可提升超过1000倍。我们提供了两种先验等价的证明。模拟研究表明，虽然两种参考先验在变量选择结果上等价，但对于大样本量，基于新型先验的计算速度可快数个数量级。最后，我们通过一项对美国本土3108个县级中位家庭收入与社会经济回归变量的空间回归研究，展示了新型参考先验的实用性。