Notation ${\displaystyle \operatorname {LKJ} (\eta )}$ ${\displaystyle \eta \in (0,\infty )}$ (shape) ${\displaystyle \Sigma }$ is a positive-definite symmetric matrix the identity matrix

In probability theory and Bayesian statistics, the Lewandowski-Kurowicka-Joe distribution, often referred to as the LKJ distribution, is a continuous probability distribution for a symmetric matrix. It is commonly used as a prior for correlation or covariance matrices in hierarchical Bayesian modelling. In hierarchical Bayesian modelling it is common to include in the model a covariance structure for the data; one of the goals of the Bayesian model will be to estimate a posterior distribution for this covariance; for this to work a prior distribution is needed for the covariance matrix, this is commonly provided by the LKJ distribution.[1] The distribution was first introduced in a more general context [2] 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 mass function for a matrix ${\displaystyle \Sigma }$ is

${\displaystyle \!f(\Sigma ;\eta )\propto [\det(\Sigma )]^{\eta -1))$

which hides the normalization factor, a complicated expression including a product over Beta functions. For ${\displaystyle \eta =1}$ the distribution is uniform over all symmetric positive definite matrices.

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. ^ Lewandowski, Daniel; Kurowicka, Dorota; Joe, Harry (2009). "Generating Random Correlation Matrices Based on Vines and Extended Onion Method". Journal of Multivariate Analysis. 100: 1989–2001.