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 counter-evidence [1]. Conservation of Expected Evidence is about both the direction of the update and its magnitude: a low probability of seeing strong evidence in one direction must be balanced by a high probability of observing weak counter-evidence in the other direction [2]. The mere expectation of encountering evidence–before you've actually seen it–should not shift your prior beliefs. It also goes by other names, including the law of total expectation and the law of iterated expectations.

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 counter-evidence). 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{matrix} P (H) & = P (H, E) + P (H, \neg E) = P (H | E) \cdot P (E) + P (H | \neg E) \cdot P (\neg E) \end{matrix}

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 counter-evidence:

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

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.

Sequences:

Filtered Evidence, Filtered Arguments by Abram Demski

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