Bartlett Couplings of the Onion and Vine LKJ Samplers

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.