Consider the following model:

$\theta \sim Gamma(\alpha_1, \beta_1)\,$

$\lambda|\theta \sim Gamma(\alpha_2, \theta)\,$

$Y|\lambda \sim Poisson(\lambda)\,$

where $\theta$, $\lambda$, and $Y$ are random variables, and $\alpha_1$, $\beta_1$, and $\alpha_2$ are constants.

Use the “inverse scale” parameterization of the gamma distribution, that is:

$f(x | \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}, \, x > 0$

Draw the above model as a Bayesian network when $Y$ is observed.

Recall that we can make inference about $\theta$ and $\lambda$ if we have a set of random samples from the joint distribution $f(\theta, \lambda | Y)\,$. Briefly explain how you would simulate random draws from this joint distribution using Gibbs sampling (“pure” Gibbs sampling, without Metropolis, which we will cover separately). Make sure you thoroughly derive and specify all distributions that you will need to sample from. Be detailed with your math.