Bayesian complexity tests I have seen an apparently interesting quantity called the Bayesian complexity, which in this case is defined as $-2(\overline{\ln\mathfrak{L}} - \ln\mathfrak{L}_\mathrm{max})$. I am not sure what this is used for. I.e., what sorts of tests can be made on this quantity and what does it imply? 
 A: I'm not totally certain about this---I've never seen that particular notation before---but that quantity looks like a variation on Deviance or the Deviance Information Criterion (DIC).  Assuming that's true, it's intended to be a goodness of fit measurement that takes "parsimony" into account, which makes it handy for model selection. 
Suppose we're trying to decide between several models. The simplistic approach would be to choose the model that minimizes the residuals (or maximizes the likelihood) of the data. However, that might lead us to choose a model that has tons of parameters and overfits the data by fitting both the underlying data as well as the noise. 
Deviance attempts to correct for this by comparing a model with a "fully saturated" model that has a parameters for all observations (I'm guessing that's what $\ln\mathfrak{L}_\mathrm{max}$ means). Models with high deviance "easily" fit the data, while models with small deviance are less likely to fit the data well by chance, so we usually prefer the model with the smaller deviance.
