Generalised Bayesian Structural Equation Modelling

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.