Consider a generic hierarchical Bayes model with data $y_i\sim p(y|\theta_i)$, dependent of parameters $\theta_i\sim p(\theta|\phi)$ and hyperparameters $\phi\sim p(\phi)$. Furthermore, assume that $\theta$ is multivariate normal with unknown mean and covariance, and thus $\phi=\{\mu,\Sigma\}$.

I have implemented in MATLAB a Gibbs sampler of the posterior distribution $p(\theta,\phi|y)$, but am unsure if the marginal posterior of $\theta$ makes sense. According to wikipedia I have decided to use a normal-inverse Wishart distribution as an hyperprior for the mean and covariance. As it can be seen in the figure below (left-side), the marginal posterior of $\theta$ seems to be tailed in the side lobes and uniform at the center, whereas the mean and covariance (right-side and below figure) have a clear and well-defined peak. Does these results make sense? I am still relatively new to conjugate priors and hierarchical Bayes models. Thank you in advance. enter image description here enter image description here


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