Suppose we have two finite partitions of the outcome space, both with the same cardinality
$H_{1}, H_{2}, ...., H_{n}$ and $E_{1}, E_{2}, ..., E_{n}$
Since these form a partition, we know that $P(E_{1}) + P(E_{2}) + .... + P(E_{n}) = 1$ and $P(E_{i}|E_{j}) = 0$ for $i \neq j$, and similarly for the $H_{i}$'s.
Suppose further that we know the probability of all the $E_{i}$'s and the conditional probabilities $P(E_{i}|H_{j})$ for all $i, j \leq n$. Is it then possible to derive the probability of the $H_{j}$'s? If so, how?