A Gibbs sampler for the LKJ Prior on correlation matrices

We propose novel Gibbs sampling methodology for fitting Bayesian mixed models of the hierarchical regression type that use the popular LKJ prior (Lewandowski, Kurowicka, and Joe, 2009) for the random-effect correlation structure. In terms of computing time and effective sample size, our Gibbs sampler is competitive with the state-of-the-art brms R package (Bürkner, 2017) that is based on Stan. Moreover, it can potentially perform much better and more reliably with sparse and modest-sized data sets that rely more heavily on the prior. The Gibbs sampling steps may utilize the recent sampling algorithm of Hamura, Irie, and Sugasawa (2024) for the multivariate generalized inverse Gaussian distribution.