The model
\[\begin{aligned} y_{ij} | \mu_j, \sigma^2 &\overset{i.i.d.}\sim \mathcal{N}(\mu_j, \sigma^2), \end{aligned}\] where \(i = 1, ..., n_j\), (number of observations in group \(j\)), and \(j = 1, ..., J\), (number of groups).
Priors
\[\begin{aligned} \mu_j | \mu, \tau^2 &\sim \mathcal{N}(\mu, \tau^2) \\ \mu | \mu_0, \tau_0^2 &\sim \mathcal{N}(\mu_0, \tau_0^2) \\ \tau^2 | \alpha_\tau, \beta_\tau &\sim \mathcal{IG}(\alpha_\tau, \beta_\tau) \Rightarrow 1/\tau^2 | \alpha_\tau, \beta_\tau \sim \mathcal{G}(\alpha_\tau, \beta_\tau) \\ \sigma^2 | \alpha_\sigma, \beta_\sigma &\sim \mathcal{IG}(\alpha_\sigma, \beta_\sigma) \Rightarrow 1/\sigma^2 | \alpha_\sigma, \beta_\sigma \sim \mathcal{G}(\alpha_\sigma, \beta_\sigma) \end{aligned}\]
Likelihood
\[\begin{aligned} L(y_{ij} | \mu_j, \mu, \sigma^2, \tau^2) &= \prod_{j=1}^J \Big\{\prod_{i = 1}^{n_j}\frac{1}{\sqrt{2 \pi \sigma^2}}exp\Big[-\frac{1}{2}\frac{(y_{ij} - \mu_j)^2}{\sigma^2}\Big]\Big\} \end{aligned}\]
Posterior
\[\begin{aligned} p(&\mu_j, \mu, \sigma^2, \tau^2 | y_{ij}, \mu_0, \tau_0^2, \alpha_\tau, \beta_\tau, \alpha_\sigma, \beta_\sigma) \propto \text{ Likelihood} \times \text{Prior} \\ \propto &L(y_{ij} | \mu_j, \mu, \sigma^2, \tau^2) \prod_{j=1}^J \Big[p(\mu_j|\mu, \tau^2)\Big]p(\mu|\mu_0, \tau_0^2)p(1/\tau^2|\alpha_\tau, \beta_\tau)p(1/\sigma^2|\alpha_\sigma, \beta_\sigma) \\ =&\prod_{j=1}^J \Big\{\prod_{i = 1}^{n_j}\frac{1}{\sqrt{2 \pi \sigma^2}}exp\Big[-\frac{1}{2}\frac{(y_{ij} - \mu_j)^2}{\sigma^2}\Big]\Big\} \\ &\prod_{j=1}^J\Big[\frac{1}{\sqrt{2 \pi \tau^2}}exp\Big[-\frac{1}{2}\frac{(\mu_j - \mu)^2}{\tau^2}\Big] \\ &\Big[\frac{1}{\sqrt{2 \pi \tau_0^2}}exp\Big[-\frac{1}{2}\frac{(\mu - \mu_0)^2}{\tau_0^2}\Big] \\ & \frac{\beta_\tau^{\alpha_\tau}}{\Gamma(\alpha_\tau)}(1/\tau^2)^{\alpha_\tau-1}exp[-\beta_\tau(1/\tau^2)] \\ & \frac{\beta_\sigma^{\alpha_\sigma}}{\Gamma(\alpha_\sigma)}(1/\sigma^2)^{\alpha_\sigma-1}exp[-\beta_\sigma(1/\sigma^2)] \end{aligned}\]
Full conditionals
\[\begin{aligned} \mu_j | . &\sim \mathcal{N}\Big(\frac{\sum_{i=1}^{n_j} y_{ij} / \sigma^2 + \mu / \tau^2}{n_j / \sigma^2 + 1/\tau^2}, \frac{1}{n_j/\sigma^2 + 1/\tau^2}\Big) \\ \mu |. &\sim \mathcal{N}\Big(\frac{\sum_{j=1}^J \mu_j/\tau^2 + \mu_0/\tau_0^2}{J/\tau^2 + 1/\tau_0^2}, \frac{1}{J/\tau^2 + 1/\tau_0^2}\Big) \\ 1/\tau^2 |. &\sim \mathcal{G}\Big(\alpha_\tau + \frac{J}{2}, \beta_\tau + \frac{\sum_{j=1}^J (\mu_j - \mu)^2}{2}\Big) \\ 1/\sigma^2 |. &\sim \mathcal{G}\Big(\alpha_\sigma + \frac{\sum_{j=1}^J n_j}{2}, \beta_\sigma + \frac{\sum_{j=1}^J \sum_{i=1}^{n_j}(y_{ij}-\mu_j)^2}{2}\Big) \end{aligned}\]
(image credit: https://stats.stackexchange.com/q/44583)