when specifying priors, for student or normal error model for instance,

usually, each parameter gets its prior separately

does it mean the posterior for mu, sig is to be understood as two independent univariate samples or a bi-variate sample, (and tri-variate for student with nu) ?

what would be the difference in the posterior, if the prior was explicitely a bivariate with a covariance parameter ?

it is legitimate to compute the empirical correlation coefficient of posterior mu and posterior sig sample for instance, having set a univariate prior for each ?

if I set a covariance parameter in a bivariate prior, then I get an entire distribution for it and not just one correlation coefficient compared to the case where each parameter has its univariate prior. so which one is better ?

If i set two univariate for mu sig, does the MCMC works considering a bivariate distribution made by the product of the priors ?

if so starting with conceptual independence between the two variables, and then as the posterior is computed, it may happen that the bivariate should be considered no longer factorisable if posterior empirical correlation coefficient is highly non zero for instance ?

in that case we can infer that prior independance is no longer an option a posteriori ?

in that latter case, shoud we restart estimation with an explicit two dimensional prior with a covariance parameter to be estimated by the model. But then his would give a three dimensional distribution. which may again show correlation, then aren’t we recursively stuck in the same problem ?