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 η {\displaystyle \eta } and the probability density function for a d × d {\displaystyle d\times d} matrix R {\displaystyle \mathbf {R} } is
p ( R ; η ) = C × [ det ( R ) ] η − 1 {\displaystyle p(\mathbf {R} ;\eta )=C\times [\det(\mathbf {R} )]^{\eta -1}}
with normalizing constant C = 2 ∑ k = 1 d − 1 ( 2 η − 2 + d − k ) ( d − k ) ∏ k = 1 d − 1 [ B ( η + ( d − k − 1 ) / 2 , η + ( d − k − 1 ) / 2 ) ] d − k {\displaystyle C=2^{\sum _{k=1}^{d-1}(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 η = 1 {\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.
Usage 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.
References
External links Described as part of the Stan manual distribution-explorer
