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Conservation of expected evidence

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Conservation of expected evidence is a theorem that says: "for every expectation of evidence, there is an equal and opposite expectation of counterevidence".

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) \cdot P(E) + P(H|\neg{E}) \cdot P(\neg{E})

But the prior probability of the hypothesis itself can be trivially broken up the same way:

\begin{alignat}{2}
P(H) & = P(H,E) + P(H,\neg{E}) \\
& = P(H|E) \cdot P(E) + P(H|\neg{E}) \cdot P(\neg{E})
\end{alignat}

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:

D_{1} = 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:

D_{1} \cdot P(E) = -D_{2} \cdot P(\neg{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.

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