I was reading the paper Bayesian linear mixed models using Stan: A tutorial for psychologists, linguists, and cognitive scientists. In section 2.3 Varying Intercepts, Varying Slopes Mixed Effects Mode, paragraph Defining a variance-covariance matrix for the random effects they suggest to define a covariance matrix to capture the correlation between the random effects (intercept and slope).
I have been playing around with random effects model (see for example previous discussion on design matrix), and I obtained nice results, without implementing the covariance matrix. The approach I took was to define two separate design matrix for the random effects (1 for intercept, 1 for slope) to have more control on the priors, I parametrized the model in non-centered way (as done in http://twiecki.github.io/blog/2017/02/08/bayesian-hierchical-non-centered/), and then sampled. The correlation between intercept and slope was not modeled directly, but it could be inferred from the posterior (intercept and slope are likely to be correlated).
I then tried to apply the method suggested in the paper, and I obtained similar results. And it seems to me that this approach is basically an alternative way to parametrize the model in non-centered way. That is, we sample from two independent distributions (1 for intercept, 1 for slope) and then we get the correlated parameters estimation with the covariance matrix (computed via the Cholesky decomposition for computational reasons).
Is this true, do I understand correctly?