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ciphergoth comments on Eutopia is Scary - Less Wrong
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To be clear, this is almost the same as the formula you gave; I'm just using the log odds ratios formulation of Bayes theorem
LOR(X|E) = LOR(X) + log(P(E|X)) - log(P(E|NOT X))
where LOR(X) = log(P(X)/P(¬X))
in other words LOR(X|E) - LOR(X) = log(P(E|X)) - log(P(E|NOT X)) the log-likelihood ratio, the weight of evidence you need to update from one to the other.