  We propose a generalised framework for Bayesian Structural Equation Modelling
(SEM) that can be applied to a variety of data types. The introduced framework
focuses on the approximate zero approach, according to which a hypothesised
structure is formulated with approximate rather than exact zero. It extends
previously suggested models by \citeA{MA12} and can handle continuous, binary,
and ordinal data. Moreover, we propose a novel model assessment paradigm aiming
to address shortcomings of posterior predictive $p-$values, which provide the
default metric of fit for Bayesian SEM. The introduced model assessment
procedure monitors the out-of-sample predictive performance of the model in
question, and draws from a list of principles to answer whether the
hypothesised theory is supported by the data. We incorporate scoring rules and
cross-validation to supplement existing model assessment metrics for Bayesian
SEM. The methodology is illustrated in continuous and categorical data examples
via simulation experiments as well as real-world applications on the `Big-5'
personality scale and the Fagerstrom test for nicotine dependence.
