Lewandowski-Kurowicka-Joe distribution

Short description: Continuous probability distribution

Lewandowski-Kurowicka-Joe distributionNotation|  LKJ⁡(η)

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Parameters|  η∈(0,∞) (shape)
Support|  𝐑 is a positive-definite matrix with unit diagonal
Mean|  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 η and the probability density function for a d×d matrix 𝐑 is

p(𝐑;η)=C×[det⁡(𝐑)]η−1

with normalizing constant C=2∑k=1d(2η−2+d−k)(d−k)∏k=1d−1[B(η+(d−k−1)/2,η+(d−k−1)/2)]d−k, a complicated expression including a product over Beta functions. For η=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. https://www.jstor.org/stable/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.

## External links

* Described as part of the Stan manual
* distribution-explorer

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