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I have the following hierachical bayesian model -
$\mathbf{x}|\mathbf{c},\sigma^2 \sim \mathcal{N}(\mathbf{x}|\mathbf{c},\sigma^2)$
$\mathbf{c}|\mathbf{c}_1,\sigma^2_2 \sim \mathcal{N}(\mathbf{c}|\mathbf{c}_1,\sigma^2_2)$
$\sigma^2 \sim \mathcal{U}(\sigma^2|r_1,r_2)$.

Here, $\mathbf{x}$ is a data sample. $r_1$, $r_2$, $c_1$ and $\sigma_2^2$ are fixed known values. $\mathcal{U}(\cdot)$ and $\mathcal{N}(\cdot)$ are uniform and normal distributions respectively. The goal is to generate samples from the posterior distribution $p(\mathbf{c},\sigma^2|\mathbf{x})$.

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