The LKJ distribution was first introduced in 2009 in a more general context [2] by Daniel Lewandowski, Dorota Kurowicka, and Harry Joe. It is an example of the vine copula, an approach to constrained high-dimensional probability distributions.
with normalizing constant , a complicated expression including a product over Beta functions. For , the distribution is uniform over the space of all correlation matrices; i.e. the space of positive definite matrices with unit diagonal.
Usage
The LKJ distribution is commonly used as a prior for correlation matrix in Bayesian hierarchical modeling. Bayesian hierarchical modeling often tries to make an inference on the covariance structure of the data, which can be decomposed into a scale vector and correlation matrix.[3] Instead of the prior on the covariance matrix such as the inverse-Wishart distribution, LKJ distribution can serve as a prior on the correlation matrix along with some suitable prior distribution on the scale vector. It has been implemented in several probabilistic programming languages, including Stan and PyMC.
![{\displaystyle p(\mathbf {R} ;\eta )=C\times [\det(\mathbf {R} )]^{\eta -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789875bdb13d8e410ffa6e3e515879a1de6d4e20)
![{\displaystyle C=2^{\sum _{k=1}^{d}(2\eta -2+d-k)(d-k)}\prod _{k=1}^{d-1}\left[B\left(\eta +(d-k-1)/2,\eta +(d-k-1)/2\right)\right]^{d-k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/291a0f65e2e811840d404932a6b586bb4d622ef8)
