Lewandowski-Kurowicka-Joe distribution

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

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]

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.

References

1. Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (2013). Bayesian Data Analysis (Third ed.). Chapman and Hall/CRC. ISBN   978-1-4398-4095-5.
2. Lewandowski, Daniel; Kurowicka, Dorota; Joe, Harry (2009). "Generating Random Correlation Matrices Based on Vines and Extended Onion Method". Journal of Multivariate Analysis. 100 (9): 1989–2001. doi: 10.1016/j.jmva.2009.04.008 .
3. Barnard, John; McCulloch, Robert; Meng, Xiao-Li (2000). "Modeling Covariance Matrices in Terms of Standard Deviations and Correlations, with Application to Shrinkage". Statistica Sinica. 10 (4): 1281–1311. ISSN   1017-0405. JSTOR   24306780.

External links

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