Bayesian Structural Equation Models of Correlation Matrices

We present a method for Bayesian structural equation modeling of sample correlation matrices as correlation structures. The method transforms the sample correlation matrix to an unbounded vector using the matrix logarithm function. Bayesian inference about the unbounded vector is performed assuming a multivariate-normal likelihood, with a mean based on the transformed model-implied correlation matrix, and a covariance assumed to be of known form. Using Monte Carlo simulation, we examine the performance of the method with normal and ordinal indicators, as well as the capacity of the method to estimate models that account for misspecification. The performance of the approach is often adequate suggesting that the proposed method can be used for Bayesian analysis of correlation structures. We conclude with a discussion of potential applications of the approach, as well as future directions needed to further develop the method.