Bayes' rule $$ \pi(\theta|y)\propto\pi(y|\theta)\pi(\theta) $$ gives the posterior distribution. However for the first time I have encountered a problem where I have two parameters (because the likelihood function is a gamma distribution), $\alpha$ and $\beta$.
- I know the likelihood function (a gamma distribution).
- I know the distribution for $\beta$ given that $\alpha$ is known (a gamma distribution).
- I know the distribution of $\alpha$ (an exponential distribution).
- I have data $y$.
The goal is to find the posterior distribution. In general, how do I approach a problem like this? How can I find the the prior when the prior is given as two separate distributions and one depends on the other?