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## 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_lupdf(y | eta).

### 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. lkj_corr_cholesky_lpdf is faster, more numerically stable, uses less memory, and should be preferred to this.

real lkj_corr_lupdf(matrix y | real eta)
The log of the LKJ density for the correlation matrix y given nonnegative shape eta dropping constant additive terms. lkj_corr_cholesky_lupdf is faster, more numerically stable, uses less memory, and should be preferred to this.

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.