Scaling WAIC for Multiple Endogenous Response Variables I'm trying to think about WAIC under a multivariate model scenario. Suppose I have one model composed of two relationships:
y1 ~ x
y2 ~ y1

This is one model. Now, I have a second model
y1 ~ x
y2 ~ y1 + x

I would like to construct a WAIC to compare the first case to the second. However, given that y1 and y2 are likely on different scales or might even share different error distributions, their WAIC values could be on radically different scales. How, then, to combine?
Intuitively, scaling WAIC values for each relationship seems like the answer. But how? Given that we're often interested in $\Delta$WAIC scores, simply centering and summing seems intuitive, but, is there a theoretical justification behind that intuition? Or is something more exotic needed?
I know this might be an odd question, but is there any literature on rescaling WAICs or perhaps likelihoods to combine multiple pieces into a more holistic score for model comparison in this type of scenario? I've been sifting through the Structural Equation Modeling literature and have not yet found something adequate to translate into a Bayesian framework.
 A: If you consider that WAIC is asymptotically same as Bayesian leave-one-out cross-validation, then you can think what is your prediction task and the cost/utility function used. Then you would not consider y1 and y2 as some arbitrary quantities, and knowing what is your prediction task would help you to choose the balance between accuracy of predicting y1 and y2. See Aki Vehtari and Janne Ojanen (2012). A survey of Bayesian predictive methods for model assessment, selection and comparison. Statistics Surveys, 6:142-228, http://dx.doi.org/10.1214/12-SS102 for more details on the decision theoretical assumptions behind WAIC and LOO-CV.
A: The WAIC, as the AIC, is scaled to the likelihood/posterior, so there should be no need to adjust it to the scales of y1 or y2. 
Moreover, as the AIC is defined point-wise, it shouldn't matter in which order you sum it up, so you can calculate WAIC(y1, y2), or also WAIC(y1) + WAIC(y2). 
Side note: blavaan is doing Bayesian SEMS based on the lavaan syntax and can calculate the WAIC. 
