Hodges James S, Carlin Bradley P, Fan Qiao
Division of Biostatistics, School of Public Health, University of Minnesota, MMC 303, 420 Delaware St. SE, Minneapolis, Minnesota 55455, USA.
Biometrics. 2003 Jun;59(2):317-22. doi: 10.1111/1541-0420.00038.
Bayesian analyses of spatial data often use a conditionally autoregressive (CAR) prior, which can be written as the kernel of an improper density that depends on a precision parameter tau that is typically unknown. To include tau in the Bayesian analysis, the kernel must be multiplied by tau(k) for some k. This article rigorously derives k = (n - I)/2 for the L2 norm CAR prior (also called a Gaussian Markov random field model) and k = n - I for the L1 norm CAR prior, where n is the number of regions and I the number of "islands" (disconnected groups of regions) in the spatial map. Since I = 1 for a spatial structure defining a connected graph, this supports Knorr-Held's (2002, in Highly Structured Stochastic Systems, 260-264) suggestion that k = (n - 1)/2 in the L2 norm case, instead of the more common k = n/2. We illustrate the practical significance of our results using a periodontal example.
空间数据的贝叶斯分析通常使用条件自回归（CAR）先验，它可以写成一个非恰当密度的核，该密度依赖于一个通常未知的精度参数τ。为了在贝叶斯分析中纳入τ，核必须乘以τ的k次方，其中k为某个值。本文严格推导了L2范数CAR先验（也称为高斯马尔可夫随机场模型）的k = (n - I)/2，以及L1范数CAR先验的k = n - I，其中n是区域的数量，I是空间地图中“岛屿”（不相连的区域组）的数量。由于对于定义连通图的空间结构，I = 1，这支持了Knorr-Held（2002年，《高度结构化随机系统》，第260 - 264页）的建议，即在L2范数情况下k = (n - 1)/2，而不是更常见的k = n/2。我们使用一个牙周病的例子来说明我们结果的实际意义。
