I am currently reading a book about mixture analysis, and in the textbook a posterior for the parameters $\mu1,\mu2,\sigma_1^2,\sigma_2^2$ of a two-component gaussian mixture is derived as follows (S and y are the classifications and data and are given):
where $p(\mu_1|\sigma_1^2,y,S)$ and $p(\mu_2|\sigma_2^2,y,S)$ are normally distributed with some parameters, which I omit for readability here, and $p(\sigma_1^2|y,S)$ and $p(\sigma_2^2|y,S)$ are inverse gamma distributed with some parameters, which I also omit for readability here.
Now my question is: The author states that this is a closed-form solution for the joint posterior $p(\mu1,\mu2,\sigma_1^2,\sigma_2^2|S,y)$. I cannot recognize this product of two subproducts as a known distribution. I also checked the Wikipedia page for conjugate priors, but I could not find anything.
Can someone tell me if I'm overlooking something here?
I already derived the full conditional distributions, so Gibbs sampling would be possible, but this is not a closed-form solution then anymore.
Thanks in advance,