# Correlation Matrix Distributions

The correlation matrix distributions have support on the (Cholesky factors of) correlation matrices. A Cholesky factor \(L\) for a \(K \times K\) correlation matrix \(\Sigma\) of dimension \(K\) has rows of unit length so that the diagonal of \(L L^{\top}\) is the unit \(K\)-vector. Even though models are usually conceptualized in terms of correlation matrices, it is better to operationalize them in terms of their Cholesky factors. If you are interested in the posterior distribution of the correlations, you can recover them in the generated quantities block via

```
generated quantities {
corr_matrix[K] Sigma;
Sigma = multiply_lower_tri_self_transpose(L); }
```

## LKJ correlation distribution

### Probability density function

For \(\eta > 0\), if \(\Sigma\) a positive-definite, symmetric matrix with unit diagonal (i.e., a correlation matrix), then \[\begin{equation*} \text{LkjCorr}(\Sigma|\eta) \propto \det \left( \Sigma \right)^{(\eta - 1)}. \end{equation*}\] 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.

### Distribution statement

`y ~`

`lkj_corr`

`(eta)`

Increment target log probability density with `lkj_corr_lupdf(y | eta)`

.

*Available since 2.3*

### 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.

*Available since 2.12*

`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.

*Available since 2.25*

`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

*Available since 2.0*

## Cholesky LKJ correlation distribution

Stan provides an implicit parameterization of the LKJ correlation matrix density in terms of its Cholesky factor, which you should use rather than the explicit parameterization in the previous section. For example, if `L`

is a Cholesky factor of a correlation matrix, then

`2.0); # implies L * L' ~ lkj_corr(2.0); L ~ lkj_corr_cholesky(`

Because Stan requires models to have support on all valid constrained parameters, `L`

will almost always^{1} be a parameter declared with the type of a Cholesky factor for a correlation matrix; for example,

`parameters { cholesky_factor_corr[K] L; # rather than corr_matrix[K] Sigma; // ... `

### Probability density function

For \(\eta > 0\), if \(L\) is a \(K \times K\) lower-triangular Cholesky factor of a symmetric positive-definite matrix with unit diagonal (i.e., a correlation matrix), then \[\begin{equation*} \text{LkjCholesky}(L|\eta) \propto \left|J\right|\det(L L^\top)^{(\eta - 1)} = \prod_{k=2}^K L_{kk}^{K-k+2\eta-2}. \end{equation*}\] See the previous section for details on interpreting the shape parameter \(\eta\). Note that even if \(\eta=1\), it is still essential to evaluate the density function because the density of \(L\) is not constant, regardless of the value of \(\eta\), even though the density of \(LL^\top\) is constant iff \(\eta=1\).

A lower triangular \(L\) is a Cholesky factor for a correlation matrix if and only if \(L_{k,k} > 0\) for \(k \in 1{:}K\) and each row \(L_k\) has unit Euclidean length.

### Distribution statement

`L ~`

`lkj_corr_cholesky`

`(eta)`

Increment target log probability density with `lkj_corr_cholesky_lupdf(L | eta)`

.

*Available since 2.4*

### Stan functions

`real`

`lkj_corr_cholesky_lpdf`

`(matrix L | real eta)`

The log of the LKJ density for the lower-triangular Cholesky factor L of a correlation matrix given shape eta

*Available since 2.12*

`real`

`lkj_corr_cholesky_lupdf`

`(matrix L | real eta)`

The log of the LKJ density for the lower-triangular Cholesky factor L of a correlation matrix given shape eta dropping constant additive terms

*Available since 2.25*

`matrix`

`lkj_corr_cholesky_rng`

`(int K, real eta)`

Generate a random Cholesky factor of a correlation matrix of order K that is distributed LKJ with shape eta; may only be used in transformed data and generated quantities blocks

*Available since 2.4*

## References

*Journal of Multivariate Analysis*100: 1989–2001.

## Footnotes

It is possible to build up a valid

`L`

within Stan, but that would then require Jacobian adjustments to imply the intended posterior.↩︎