In chapter 5.4 in the book Bayesian Data Analysis by Gelman et al. I see the following expression related to a hierarchical model: $$p(\mu,\tau|y) \propto p(\mu,\tau)p(y|\mu,\tau)$$ How do I derive this expression?
1 Answer
$\begingroup$
$\endgroup$
Bayes rule says that: $$p(\mu,\tau|y)=\frac{p(y|\mu,\tau)p(\mu,\tau)}{p(y)}$$
which means $p(\mu,\tau|y)\propto p(y|\mu,\tau)p(\mu,\tau)$, i.e. LHS is proportional to the RHS.
This is typically done in Bayesian analyses where you're interested in the form of the posterior distribution and get rid of the constant part that doesn't change wrt the RVs of interest. For example, a subsequent analysis might be interested in calculating MAP estimates for $\mu,\tau$ by maximizing the posterior, in which case $p(y)$ has no effect.
