基于信息距离的相关矩阵先验分布，以目标参考为中心
Informative Distance-Based Priors for Correlation Matrices Centred on a Target Reference
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中文总结 AI 辅助
研究在贝叶斯分析中指定相关矩阵先验分布的挑战及解决方法，提出基于距离的先验分布，通过费希尔弧长距离分配质量，引入参数化适应条件独立约束，提供算法实现先验预测检查和敏感性分析。
中文摘要 AI 辅助
在贝叶斯分析中，指定相关矩阵空间上的先验分布一直是个挑战。该空间是一个弯曲流形，其维度随变量数量二次增长，难以编码实质性先验信念。我们提出一种基于距离的先验分布，它根据与用户指定参考相关矩阵的费希尔弧长距离以指数方式分配质量，能向任何目标相关结构收缩。形式上，它被构建为惩罚复杂度先验，但解释有所不同。为适应条件独立约束，引入一种参数化方法，通过相对于用户提供图的逆相关矩阵的乔列斯基因子构建相关矩阵。该先验对于其速率参数的每个正值都是恰当的，适用于任何图结构下的正负相关，并在图完整时简化为完全无结构先验。还提供了直接采样算法，可在“graphpcor”包中实现先验预测检查和敏感性分析。
英文摘要
Specifying a prior over the space of correlation matrices is a persistent challenge in Bayesian analysis. The space is a curved manifold whose dimension grows quadratically with the number of variables, making substantive prior beliefs difficult to encode.\\ We propose a distance-based prior that assigns mass decaying exponentially in the Fisher arc-length distance from a user-specified reference correlation matrix, enabling shrinkage toward any target correlation structure rather than being confined to the identity matrix. Formally, this is constructed as a Penalised Complexity prior, but its interpretation shifts accordingly: unless the chosen target represents a structurally simpler state, the shrinkage penalises deviation rather than complexity in the usual sense. To accommodate conditional independence constraints, we introduce a parameterisation that constructs the correlation matrix via the Cholesky factor of the inverse correlation matrix with respect to a user-supplied graph, thereby reducing the number of free parameters from one per variable pair to one per graph edge. The prior is proper for every positive value of its rate parameter, accommodates correlations of either sign under any graph structure, and reduces to a fully unstructured prior when the graph is complete. A direct sampling algorithm is provided, enabling prior predictive checks and sensitivity analysis, implemented within the \texttt{graphpcor} package.
