# Conservation of expected evidence

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:

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

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

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

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:

${\displaystyle D_{1}=P(H|E)-P(H)\,}$

If the evidence is negative, the change is:

${\displaystyle D_{2}=P(H|\neg {E})-P(H)\ }$

Expectation of the change given positive evidence is equal to negated expectation of the change given counterevidence:

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

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.