Lewandowski-Kurowicka-Joe distribution

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] It is commonly used as a prior for correlation matrix in hierarchical Bayesian modeling. Hierarchical Bayesian 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.^[2] 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. The distribution was first introduced in a more general context ^[3] and is an example of the vine copula, an approach to constrained high-dimensional probability distributions. It has been implemented as part of the Stan probabilistic programming language and as a library linked to the Turing.jl probabilistic programming library in Julia.

The distribution has a single shape parameter [math]\displaystyle{ \eta }[/math] and the probability density function for a [math]\displaystyle{ d\times d }[/math] matrix [math]\displaystyle{ \mathbf{R} }[/math] is

[math]\displaystyle{ p(\mathbf{R}; \eta) = C \times [\det(\mathbf{R})]^{\eta-1} }[/math]

with normalizing constant [math]\displaystyle{ C=2^{\sum_{k=1}^d (2\eta - 2 +d - k)(d-k)}\prod_{k=1}^{d-1}\left[B\left(\eta + (d-k-1)/2, \eta + (d-k-1)/2\right)\right]^{d-k} }[/math], a complicated expression including a product over Beta functions. For [math]\displaystyle{ \eta=1 }[/math], the distribution is uniform over the space of all correlation matrices; i.e. the space of positive definite matrices with unit diagonal.

References

External links

• Described as part of the Stan manual
• distribution-explorer

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