Is posterior distribution really multidimensionnal?

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 ?

These are a lot of good questions. But they suggest that you really need to get a book on Bayesian analysis before you start trying to use Stan. Here are some brief answers:

Even if the priors are independent, the posterior distribution conditional on the data is always joint.

Yes, but how much depends on the model and the data.

Yes, although it might be an underestimate if the true correlation is large but your prior assumes independence.

Generally, having dependence in the prior is better but it is fine to start with independent priors.

Yes, but the bivariate distribution reflects the independence assumption.

Correct, but that doesn’t mean you should go back an reestimate the model. But people often start with independent priors, and then condition on a lot of data, and look at the posterior distribution, which has dependence. In some cases, they just stop. In other cases, they intended to go back and estimate a model with a joint prior but were just starting with a simpler model to get it going.

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