The following is from a textbook one bayesian stats. that I can't understand some deduction. It is relevant about multiple parameters to be estimated.
The jth observation in the ith group is denoted by $y_{ij}$,
where
$(y_{ij}|\mu_i,\sigma)\sim N(\mu_i,\sigma^2) \quad j=1,2, \dots, n_i \quad i= 1,2, \dots, m$
Also the $y_{ij}$ are independent from each other.
Suppose $\mu_i \sim N(\mu,\tau^2)$ and denote
$\theta= (\mu, \log(\sigma),\log(\tau))$ $Y=\{y_{ij}: j=1,\dots, n_i, i=1,\dots, n\}$ $Z=(\mu_1,\dots, \mu_m)$ $n=n_1+n_2+\cdots +n_m$
So $\theta$ is the unknown parameters interested. Take its prior distribution as $p(\theta) \propto \tau$. Then by Bayes rule, it is not difficult to get the posterior distribution:
$p(Z,\theta|Y) \propto p(\theta) \prod\limits_{i = 1}^m {p(\mu_i|\mu,\tau)} \prod\limits_{i = 1}^m \prod\limits_{j = 1}^{n_i} {p(y_{ij}|\mu_i,\sigma)}$
This is the place I can't understand. How to get this formula if No.3 formula is not correct in this thread: I am confused about Bayes' rule in MCMC
Could someone explain it in detail? If there are any excellent books that could help me, please list them.