Lewandowski-Kurowicka-Joe distribution
| Notation | [math]\displaystyle{ \operatorname{LKJ}(\eta) }[/math] | ||
|---|---|---|---|
| Parameters | [math]\displaystyle{ \eta\in (0, \infty) }[/math] (shape) | ||
| Support | [math]\displaystyle{ \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 symmetric matrices with unit diagonals.[1] It 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.[2] 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. The distribution was first introduced in a more general context [3] and is an example of the vine copula, an approach to constrained high-dimensional probability distributions. 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.
The distribution has a single shape parameter [math]\displaystyle{ \eta }[/math] and the probability density function for a [math]\displaystyle{ d\times d }[/math] matrix [math]\displaystyle{ \mathbf{R} }[/math] is
- [math]\displaystyle{ p(\mathbf{R}; \eta) = C \times [\det(\mathbf{R})]^{\eta-1} }[/math]
with normalizing constant [math]\displaystyle{ 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} }[/math], a complicated expression including a product over Beta functions. For [math]\displaystyle{ \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.
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.
- ↑ 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. https://www.jstor.org/stable/24306780.
- ↑ 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.
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