Trying to derive the posterior distribution for the following model with $n$ observations $$y=\beta_{0} + \beta_{1}X_{1}+\beta_{2}X_{2}+\epsilon$$
where the error terms $\epsilon$ follow a normal distribution $N(0,\sigma^2_{\epsilon})$ with a normal prior on $\beta_0 \sim N(0, \sigma^2_{\beta_0})$ and normal priors on coefficient terms $\beta_1, \beta_2 \sim N(0, \sigma^2_{\beta})$.
$\sigma^2_{\epsilon}$ has an inverse gamma prior with parameters $\alpha, \gamma$.
$\sigma^2_{\beta_0} \sigma^2_{\beta} \alpha, \gamma$ are constant.
Im having particular trouble defining the likelihood. Any help much appreciated!