24.1 LKJ Correlation Distribution
24.1.1 Probability Density Function
For \(\eta > 0\), if \(\Sigma\) a positive-definite, symmetric matrix with unit diagonal (i.e., a correlation matrix), then \[ \text{LkjCorr}(\Sigma|\eta) \propto \det \left( \Sigma \right)^{(\eta - 1)}. \] The expectation is the identity matrix for any positive value of the shape parameter \(\eta\), which can be interpreted like the shape parameter of a symmetric beta distribution:
if \(\eta = 1\), then the density is uniform over correlation matrices of order \(K\);
if \(\eta > 1\), the identity matrix is the modal correlation matrix, with a sharper peak in the density at the identity matrix for larger \(\eta\); and
for \(0 < \eta < 1\), the density has a trough at the identity matrix.
if \(\eta\) were an unknown parameter, the Jeffreys prior is proportional to \(\sqrt{2\sum_{k=1}^{K-1}\left( \psi_1\left(\eta+\frac{K-k-1}{2}\right) - 2\psi_1\left(2\eta+K-k-1 \right)\right)}\), where \(\psi_1()\) is the trigamma function
See (Lewandowski, Kurowicka, and Joe 2009) for definitions. However, it is much better computationally to work directly with the Cholesky factor of \(\Sigma\), so this distribution should never be explicitly used in practice.
24.1.2 Sampling Statement
y ~ lkj_corr(eta)
Increment target log probability density with lkj_corr_lpdf(y | eta)
dropping constant additive terms.
24.1.3 Stan Functions
real lkj_corr_lpdf(matrix y | real eta)
The log of the LKJ density for the correlation matrix y given
nonnegative shape eta. The only reason to use this density function is
if you want the code to run slower and consume more memory with more
risk of numerical errors. Use its Cholesky factor as described in the
next section.
matrix lkj_corr_rng(int K, real eta)
Generate a LKJ random correlation matrix of order K with shape eta;
may only be used in transformed data and generated quantities blocks
References
Lewandowski, Daniel, Dorota Kurowicka, and Harry Joe. 2009. “Generating Random Correlation Matrices Based on Vines and Extended Onion Method.” Journal of Multivariate Analysis 100: 1989–2001.