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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.

Accounting for model misspecification in Bayesian structural equation models is an active area of research. We present a uniquely Bayesian approach to misspecification that models the degree of misspecification as a parameter--a parameter akin to the correlation root mean squared residual. The misspecification parameter can be interpreted on its own terms as a measure of absolute model fit and allows for comparing different models fit to the same data. By estimating the degree of misspecification simultaneously with structural parameters, the uncertainty about structural parameters reflects the degree of model misspecification. This results in a model that produces more reliable inference than extant Bayesian structural equation modeling. In addition, the approach estimates the residual covariance matrix that can be the basis for diagnosing misspecifications and updating a hypothesized model. These features are confirmed using simulation studies. Demonstrations with a variety of real-world examples show additional properties of the approach.