AI 中文总结
该研究将洋葱与C-vine LKJ采样器归为Wang等人提出的受限Wishart表示的行归一化Bartlett构造，提出的采样器效率更高，适用于所有η>0，仅需标准正态和卡方变量。
AI 中文摘要
Lewandowski、Kurowicka和Joe（2009）提出的扩展洋葱（extended-onion）与C-vine构造是从相关矩阵上的LKJ_n(η)分布采样的标准方法。我们证明这两种构造均源自Wang、Wu和Chu（2018）提出的受限Wishart表示相关的更简单的行归一化Bartlett构造，该构造可复用经典采样器会重新生成的随机量。两个精确的逐行耦合证实了这一点：提供洋葱方向的高斯向量的平方范数恰好具有Beta半径某一分量所需的Gamma分布，而同一向量结合一个卡方变量，可生成整个行的相互独立的C-vine偏相关，且符合其所需的对称Beta分布。在Gamma-ratio统计下，归一化Bartlett、洋葱、传统对称Beta C-vine分别需要n-1、2(n-1)、n(n-1)个等价Gamma调用。受控基准测试证实，该采样器在低维场景下优于洋葱实现，且在测试的C-vine实现中持续保持优势；直接Bartlett归一化还可避免减法补运算，将小η下的零对角线阈值从机器精度尺度移至亚正常范围。该采样器对所有实数η>0均有效，且仅需标准正态和卡方变量。
英文摘要
The extended-onion and C-vine constructions of Lewandowski, Kurowicka and Joe (2009) are standard methods for sampling from the LKJ_n(eta) distribution on correlation matrices. We show that both arise from the simpler row-normalized Bartlett construction associated with the restricted-Wishart representation of Wang, Wu and Chu (2018), which exposes redundancies hidden in standard gamma-based implementations of the classical constructions. Two exact row-wise couplings establish this: the squared norm of the Gaussian vector supplying the onion's direction has exactly the Gamma law required for one component of the Beta radius, and the same vector, with one chi-squared variate, generates the entire row of mutually independent C-vine partial correlations with their required symmetric-Beta laws. We also show that, relative to flat off-diagonal measure, the LKJ family maximizes entropy at fixed expected log-determinant; the dual natural parameter is eta-1. Under gamma-ratio accounting, normalized Bartlett, the onion, and the conventional symmetric-Beta C-vine require n-1, 2(n-1), and n(n-1) Gamma-equivalent calls. Controlled benchmarks confirm a low-dimensional advantage over the onion implementation and a persistent advantage over the C-vine implementations examined; direct Bartlett normalization also avoids subtractive complements, moving the small-eta zero-diagonal threshold from machine-epsilon scale toward the subnormal range. The sampler is valid for every real eta > 0 and requires only standard normal and chi-squared variates.
CommentsCode and results: https://github.com/reinhardhansen/BartlettLKJ (archived at doi:10.5281/zenodo.22088155)
