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]

Lewandowski-Kurowicka-Joe distribution

Notation	${\displaystyle \operatorname {LKJ} (\eta )}$
Parameters	${\displaystyle \eta \in (0,\infty )}$ (shape)
Support	${\displaystyle \mathbf {R} }$ is a positive-definite matrix with unit diagonal
Mean	the identity matrix

Introduction

The LKJ distribution was first introduced in 2009 in a more general context ^[2] 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 ${\displaystyle \eta }$ and the probability density function for a ${\displaystyle d\times d}$ matrix ${\displaystyle \mathbf {R} }$ is

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

with normalizing constant ${\displaystyle C=2^{\sum _{k=1}^{d-1}(2\eta -2+d-k)(d-k)}\prod _{k=1}^{d-1}\left[B\left(\eta +(d-k-1)/2,\eta +(d-k-1)/2\right)\right]^{d-k}}$, a complicated expression including a product over Beta functions. For ${\displaystyle \eta =1}$, 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 Bayesian hierarchical modeling. Bayesian hierarchical 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.^[3] 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 in several probabilistic programming languages, including Stan and PyMC.