The Lewandowski-Kurowicka-Joe distribution, or LKJ distribution, is a statistical distribution on real symmetric positive-definite
matrices with unit diagonal. Such matrices are correlation matrices. For a 
correlation matrix 
, its probability
density function with respect to Lebesgue measure
on the 
free off-diagonal entries has the form
 |
where 
.
This Lebesgue measure treats the free entries
as ordinary coordinates in 
, restricted to the region for which 
is positive definite.
The case 
is the uniform distribution on this region
relative to that measure, meaning its density is constant there. It does not make
the off-diagonal entries independent, since they must jointly form a correlation
matrix. Values 
favor correlation matrices near the identity
matrix, while 
puts more weight near the boundary, where limiting correlation
matrices are singular. The LKJ distribution
is widely used in Bayesian models of correlation
matrices.
