Notation ${\displaystyle \operatorname {LKJ} (\eta )}$ ${\displaystyle \eta \in (0,\infty )}$ (shape) ${\displaystyle \mathbf {R} }$ is a positive-definite matrix with unit diagonal the identity matrix

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.[1] 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.[2] 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 [3] 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 ${\displaystyle \eta }$ and the probability density function for a ${\displaystyle d\times d}$ matrix ${\displaystyle \mathbf {R} }$ is

${\displaystyle p(\mathbf {R} ;\eta )=C\times [\det(\mathbf {R} )]^{\eta -1))$

with normalizing constant ${\displaystyle C=2^{\sum _{k=1}^{d}(2\eta -2+d-k)(d-k)}\prod _{k=1}^{d-1}\left[B\left(\eta +(d-k-1)/2,\eta +(d-k-1)/2\right)\right]^{d-k))$, a complicated expression including a product over Beta functions. For ${\displaystyle \eta =1}$, the distribution is uniform over the space of all correlation matrices; i.e. the space of positive definite matrices with unit diagonal.

## References

1. ^ Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (2013). Bayesian Data Analysis (Third ed.). Chapman and Hall/CRC. ISBN 978-1-4398-4095-5.
2. ^ Barnard, John; McCulloch, Robert; Meng, Xiao-Li (2000). "Modeling Covariance Matrices in Terms of Standard Deviations and Correlations, with Application to Shrinkage". Statistica Sinica. 10 (4): 1281–1311. ISSN 1017-0405. JSTOR 24306780.
3. ^ Lewandowski, Daniel; Kurowicka, Dorota; Joe, Harry (2009). "Generating Random Correlation Matrices Based on Vines and Extended Onion Method". Journal of Multivariate Analysis. 100 (9): 1989–2001. doi:10.1016/j.jmva.2009.04.008.