Let's say I have a Bayesian network with both numeric and categorical variables. I run several MCMC chains to collect samples from the distribution. Now, if the chains are "similar enough" after some number of samples, it is reasonable to assume that they mixed.
The problem is this "similar enough" thingy. In literature I always get blocked by notions like "autocovariance" and "variance", which apply to numerical variables only. What to do when the network contains also categorical variables?
One idea would be to use some statistical distance applied to empirical distributions produced by the chains. If the mutual distances are small enough, then declare victory. The problem with this approach is that again the distance 'measures' are tailored either to categorical probability distributions (Kullback–Leibler divergence) or to numerical distributions (Kolmogorov–Smirnov statistic).
Is there some other approach?