We have three events $A$, $B$ and $C$ in question, and given appropriate priors, we derive the posterior $\Pr(A|B)$. Now we want to derive a 'second-order' posterior $\Pr(A|B,C)$ by using the 'first-order' posterior $\Pr(A|B)$ as the prior.
First of all, does this mean that $\Pr(A|B,C)$ is the same as $\Pr((A|B)|C)$? If so, is the following correct:
and how do we derive $\Pr(C|(A|B))$?