  Accounting for the complexity of psychological theories requires methods that
can predict not only changes in the means of latent variables -- such as
personality factors, creativity, or intelligence -- but also changes in their
variances. Structural equation modeling (SEM) is the framework of choice for
analyzing complex relationships among latent variables, but current methods do
not allow modeling latent variances as a function of other latent variables. In
this paper, we develop a Bayesian framework for Gaussian distributional SEM
which overcomes this limitation. We validate our framework using extensive
simulations, which demonstrate that the new models produce reliable statistical
inference and can be computed with sufficient efficiency for practical everyday
use. We illustrate our framework's applicability in a real-world case study
that addresses a substantive hypothesis from personality psychology.
