I am reading the book Doing Bayesian Data Analysis by John K. Kruschke. In the chapter related to Gibbs sampling, the author suggests that we cannot understand the difference of two parameters by looking at their posterior distributions and that we should analyze the posterior of the parameters difference.
He also provides an example with two cases: θ1 and θ2 have the same posterior distributions across both cases but in the first one the parameters are positively correlated while in the second one they are negatively correlated. Then in the first case the distribution of the difference in narrow and far from 0 while in the second one it is much wider.
I always thought that there was no difference between sampling from two parameters posterior (and then taking the difference of the traces) and directly sampling from the difference posterior.
Are there cases in which the posteriors difference is not the same posterior of the difference?