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 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.

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. 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.