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.

Introduction

The LKJ distribution was first introduced in 2009 in a more general context by Daniel Lewandowski, Dorota Kurowicka, and Harry Joe. It is an example of the vine copula, an approach to constrained high-dimensional probability distributions.

The distribution has a single shape parameter $\eta$ and the probability density function for a $d\times d$ matrix $\mathbf$ is
:p(\mathbf; \eta) = C \times det(\mathbf)
with normalizing constant C=2^\prod_^\left \left(\eta + (d-k-1)/2, \eta + (d-k-1)/2\right)\right, a complicated expression including a product over Beta functions. For $\eta=1$, the distribution is uniform over the space of all correlation matrices; i.e. the space of positive definite matrices with unit diagonal.

Usage

The LKJ distribution 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. 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. 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.

References

External links

Described as part of the Stan manual

Random matrices
Bayesian statistics
Continuous distributions
Multivariate continuous distributions

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