Lewandowski-Kurowicka-Joe distribution

Continuous probability distribution

Lewandowski-Kurowicka-Joe distribution
Notation	$\operatorname {LKJ} (\eta )$
Parameters	$\eta \in (0,\infty )$ (shape)
Support	$\mathbf {R}$ 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 $\eta$ and the probability density function for a $d\times d$ matrix $\mathbf {R}$ is

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

with normalizing constant $C=2^{\sum _{k=1}^{d}(2\eta -2+d-k)(d-k)}\prod _{k=1}^{d-1}\left[B\left(\eta +(d-k-1)/2,\eta +(d-k-1)/2\right)\right]^{d-k}$ , 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.^[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 as part of the Stan probabilistic programming language and as a library linked to the Turing.jl probabilistic programming library in Julia.

References

^ 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.
^ 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.
^ 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

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda