Conservation of Expected Evidence is a consequence of probability theory which states that for every expectation of evidence, there is an equal and opposite expectation of counterevidence. [1] The mere expectation of encountering evidence–before you've actually seen it–should not shift your prior beliefs.
A consequence of this principle is that absence of evidence is evidence of absence.
Consider a hypothesis H and evidence (observation) E. Prior probability of the hypothesis is P(H); posterior probability is either P(H|E) or P(H|¬E), depending on whether you observe E or not-E (evidence or counterevidence). The probability of observing E is P(E), and probability of observing not-E is P(¬E). Thus, expected value of the posterior probability of the hypothesis is:
P(H|E) ⋅ P(E) + P(H|¬E) ⋅ P(¬E)
But the prior probability of the hypothesis itself can be trivially broken up the same way:
Thus, expectation of posterior probability is equal to the prior probability.
In other way, if you expect the probability of a hypothesis to change as a result of observing some evidence, the amount of this change if the evidence is positive is:
D1 = P(H|E) − P(H)
If the evidence is negative, the change is:
\(D_{2} = P(H|\neg{E})-P(H)\\)
Expectation of the change given positive evidence is equal to negated expectation of the change given counterevidence:
D1 ⋅ P(E) = − D2 ⋅ P(¬E)
If you can anticipate in advance updating your belief in a particular direction, then you should just go ahead and update now. Once you know your destination, you are already there. On pain of paradox, a low probability of seeing strong evidence in one direction must be balanced by a high probability of observing weak counterevidence in the other direction.