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. It 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. The distribution was first introduced in a more general context  and is an example of the vine copula, an approach to constrained high-dimensional probability distributions. 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. The distribution has a single shape parameter  and the probability density function for a  matrix  is

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. References

External links 

 Described as part of the Stan manual
 distribution-explorer

Random matrices
Bayesian statistics
Continuous distributions
Multivariate continuous distributions