[Paper Review] On equivalence of the LKJ distribution and the restricted Wishart distribution

Zhenxun Wang, Yunan Wu+1 more|arXiv (Cornell University)|Sep 13, 2018

Advanced Statistical Methods and Models4 citations

TL;DR

This paper establishes the equivalence between the restricted Wishart distribution and the LKJ distribution, demonstrating that both yield the same prior over correlation matrices. The authors propose a new method to generate random correlation matrices via the restricted Wishart distribution, which is significantly faster than the traditional onion method—especially in low dimensions (T < 120), with speedups up to 2.2× in simulations.

ABSTRACT

In this paper, we want to show the Restricted Wishart distribution is equivalent to the LKJ distribution, which is one way to specify a uniform distribution from the space of positive definite correlation matrices. Based on this theorem, we propose a new method to generate random correlation matrices from the LKJ distribution. This new method is faster than the original onion method for generating random matrices, especially in the low dimension ($T&lt;120$) situation.

Motivation & Objective

• To establish theoretical equivalence between the restricted Wishart distribution and the LKJ distribution for correlation matrices.
• To provide a new computational method for generating random correlation matrices from the LKJ distribution using the restricted Wishart framework.
• To improve computational efficiency in Bayesian modeling by offering a faster alternative to the existing onion method.
• To deepen understanding of the Wishart distribution through partial correlation parameterization.

Proposed method

• The restricted Wishart distribution is derived by applying the separation strategy to the Wishart distribution, decomposing the covariance matrix into variances and a correlation matrix.
• The method uses the Bartlett decomposition to generate a Wishart-distributed matrix via lower-triangular matrix factorization with chi-squared and standard normal variates.
• The joint density of the correlation matrix under the restricted Wishart distribution is derived and shown to match the LKJ density through Jacobian transformation and gamma function identities.
• The equivalence is proven using mathematical induction on the normalization constant, confirming that the restricted Wishart distribution is identical in law to the LKJ distribution.
• Random correlation matrices are generated by sampling from the restricted Wishart distribution and transforming the resulting covariance matrix into a correlation matrix.
• Computational performance is benchmarked against the onion method using R, with timing measured across dimensions T = 20 to 280.

Experimental results

Research questions

• RQ1 Is the restricted Wishart distribution mathematically equivalent to the LKJ distribution for correlation matrices?
• RQ2 Can the restricted Wishart distribution be used as an efficient alternative to the onion method for generating LKJ-distributed correlation matrices?
• RQ3 What is the computational performance advantage of the restricted Wishart method over the onion method in low- and high-dimensional settings?
• RQ4 How does the partial correlation parameterization in the restricted Wishart framework relate to the LKJ distribution's construction?

Key findings

• The restricted Wishart distribution is mathematically equivalent to the LKJ distribution, as proven via normalization constant verification using induction and gamma function identities.
• The new method based on the restricted Wishart distribution generates random correlation matrices faster than the onion method, particularly in low dimensions (T < 120), with a 2.2× speedup observed at T = 20.
• At T = 20, the restricted Wishart method required 0.70 seconds to generate 5,000 matrices, compared to 1.53 seconds for the onion method.
• For higher dimensions (T ≥ 240), the performance gap narrows, with both methods taking approximately 44–50 seconds for 5,000 matrices.
• The restricted Wishart and inverse restricted Wishart methods exhibit comparable computational efficiency, with both outperforming the onion method in low dimensions.
• The equivalence provides theoretical justification for using the Wishart distribution as a prior on correlation matrices through partial correlation parameterization.