The standard assumption about updating for sets of probabilities is that your degree of belief in \(X\) after learning \(E\) is given by \(P(X\mid E) = \{p(X\mid E), p\in P, p(E) &gt; 0\}\). Your belief state after having learned \(E\) is \(P(\cdot\mid E) = \{p(\cdot\mid E), p\in P, p(E) &gt; 0\}\). That is, by the set of conditional probabilities.