# Bayesian Gamma Regression Update

I'm looking for a resource that explains how to do update the coefficients for a Bayesian gamma regression using Gibbs sampling. Specifically, if

$$y_i \sim Gamma(\alpha,\beta_i)$$ and my data likelihood is $$\prod_{i=1}^NP(y_i | \alpha,\beta_i)$$

$$log(\mu_i) = x_i \gamma_1 + \gamma_0$$

$$\alpha = \mu_i^2 / \phi$$

$$\beta_i = \mu_i/\phi$$

assuming the shape parameter $$\alpha$$ is a known constant, I need to sample $$\gamma_1$$ and $$\gamma_0$$ from the full conditionals. For example, $$P(\gamma_1 | \gamma_0,Y,X,\alpha) \propto P(y_i | \alpha, \alpha /exp(x_i \gamma_1 + \gamma_0))P(\gamma_1|\mu_0,\sigma^2_0)$$.

However, the first term is gamma density and second is normal and are thus not conjugate. What is the best approach to sampling from this conditional? I can think of three options but I'm not sure what's best in terms of convergence time.

1) Metropolis proposals for $$\gamma_1$$. Proposal width can be problematic.