Lewandowski-Kurowicka-Joe distribution

In probability theory and Bayesian statistics, the Lewandowski-Kurowicka-Joe distribution, often referred to as the LKJ distribution, is a probability distribution over positive definite symmetric matrices with unit diagonals.

Introduction
The LKJ distribution was first introduced in 2009 in a more general context by Daniel Lewandowski, Dorota Kurowicka, and Harry Joe. It is an example of the vine copula, an approach to constrained high-dimensional probability distributions.

The distribution has a single shape parameter $$\eta$$ and the probability density function for a $$d\times d$$ matrix $$\mathbf{R}$$ is
 * $$p(\mathbf{R}; \eta) = C \times [\det(\mathbf{R})]^{\eta-1}$$

with normalizing constant $$C=2^{\sum_{k=1}^d (2\eta - 2 +d - k)(d-k)}\prod_{k=1}^{d-1}\left[B\left(\eta + (d-k-1)/2, \eta + (d-k-1)/2\right)\right]^{d-k}$$, a complicated expression including a product over Beta functions. For $$\eta=1$$, 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 hierarchical Bayesian modeling. Hierarchical Bayesian 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. 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 as part of the Stan probabilistic programming language and as a library linked to the Turing.jl probabilistic programming library in Julia.