| Lewandowski-Kurowicka-Joe distribution | |
| Type: | mass |
| Notation: | \operatorname{LKJ}(η) |
| Parameters: | η\in(0,infty) |
| Support: | R |
| 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]
The LKJ distribution was first introduced in 2009 in a more general context [2] 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
d x d
R
p(R;η)=C x [\det(R)]η-1
| ||||||||||
| C=2 |
| d-1 | |
| \prod | |
| k=1 |
\left[B\left(η+(d-k-1)/2,η+(d-k-1)/2\right)\right]d-k
η=1
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.[3] 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.