View source for Lewandowski-Kurowicka-Joe distribution

{{short description|Continuous probability distribution}}
{{Infobox probability distribution
  | name       = Lewandowski-Kurowicka-Joe distribution
  | type       = mass
  | notation   = <math>\operatorname{LKJ}(\eta)</math>
  | parameters = <math>\eta\in (0, \infty) </math>  (shape)
  | support    = <math>\mathbf{R}</math> 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 matrix|positive definite]] symmetric matrices with unit diagonals.<ref name="bda">{{Cite book |last1=Gelman |first1=Andrew |title=Bayesian Data Analysis |last2=Carlin |first2=John B. |last3=Stern |first3=Hal S. |last4=Dunson |first4=David B. |last5=Vehtari |first5=Aki |last6=Rubin |first6=Donald B. |publisher=Chapman and Hall/CRC |year=2013 |isbn=978-1-4398-4095-5 |edition=Third |author-link1=Andrew Gelman |author-link2=John Carlin (professor) |author-link6=Donald Rubin}}</ref>

== Introduction ==
The LKJ distribution was first introduced in 2009 in a more general context <ref name="lkj">{{Cite journal|last1=Lewandowski|first1=Daniel|last2=Kurowicka|first2=Dorota|last3=Joe|first3=Harry|year=2009|title=Generating Random Correlation Matrices Based on Vines and Extended Onion Method|journal=Journal of Multivariate Analysis|volume=100|issue=9 |pages=1989–2001|doi=10.1016/j.jmva.2009.04.008 |doi-access=free}}</ref> 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 <math>\eta</math> and the [[probability density function]] for a <math>d\times d</math> matrix <math>\mathbf{R}</math> is
:<math>p(\mathbf{R}; \eta) = C \times [\det(\mathbf{R})]^{\eta-1}</math>
with normalizing constant <math>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}</math>, a complicated expression including a product over [[Beta function]]s. For <math>\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.

== 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.<ref>{{Cite journal |last1=Barnard |first1=John |last2=McCulloch |first2=Robert |last3=Meng |first3=Xiao-Li |title=Modeling Covariance Matrices in Terms of Standard Deviations and Correlations, with Application to Shrinkage |date=2000 |url=https://www.jstor.org/stable/24306780 |journal=Statistica Sinica |volume=10 |issue=4 |pages=1281–1311 |jstor=24306780 |issn=1017-0405}}</ref> 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 language]]s, including [[Stan (software)|Stan]] and [[PyMC]].

== References == 
{{Reflist}}

== External links ==

* [https://mc-stan.org/docs/functions-reference/lkj-correlation.html Described as part of the Stan manual]
* [https://distribution-explorer.github.io/multivariate_continuous/lkj.html distribution-explorer]

{{ProbDistributions|multivariate}}

[[Category:Random matrices]]
[[Category:Bayesian statistics]]
[[Category:Continuous distributions]]
[[Category:Multivariate continuous distributions]]




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