Model:
The following model corresponds to samples drawn from a Gaussian distribution with unknown mean and unknown variance:
\begin{align}
x | \mu, \sigma^2 &\sim \mathcal{N}(\mu, \sigma^2 )\\
\mu | \mu_0, \sigma_0^2 &\sim \mathcal{N}(\mu_0, \sigma_0^2)\\
\sigma^2 | \alpha, \beta &\sim Inverse Gamma(\alpha, \beta)
\end{align}
Task:
I want to infer both $\mu$ and $\sigma^2$
Inference:
The conditionals are: \begin{align} p(\mu | \sigma^2, x) &\propto_\mu p(x | \mu, \sigma^2) p(\mu | \mu_0, \sigma_0^2)\\ p(\sigma^2 | \mu) &\propto_{\sigma^2} p(x | \mu, \sigma^2) p(\sigma^2 | \alpha, \beta) \end{align}
Since in the upper equation we have two Normals (which are conjugate with respect to $\mu$), we can easily get the conditional, which is a Normal distribution.
Since in the lower equation we have a normal and an Inverse Gamma (which are conjugate with respect to $\sigma^2$), we can easily get the conditional, which is and Inverse Gamma.
And since we can get the conditional, we can Gibbs sample them both to get their marginal posteriors $p(\mu | x) $ and $p(\sigma^2 | x)$.
Question:
I see that to avoid losing conjugacy $1/\sigma_0$ is forced to be $\rho/\sigma$ (see Michael Jordan's notes or Rasmussen's paper on DP-GMM which I am trying to implement).
But why are we losing conjugacy? Why can't I do this Gibbs sampling using the conditionals above? What am I missing?
EDIT:
I get that the joint posterior of $\mu, \sigma^2$: \begin{align} p(\mu, \sigma^2 | x) \propto p(\mu | \mu_0, \sigma_0) p(\sigma^2 | \alpha, \beta) p (x | \mu, \sigma) \end{align}
cannot be computed from the product of the two conditional posteriors above. But I wonder whether I should use this joint posterior instead of the individual posteriors explained above.