**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)⋅P(E)+P(H|¬E)⋅P(¬E)

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

P(H)=P(H,E)+P(H,¬E)=P(H|E)⋅P(E)+P(H|¬E)⋅P(¬E)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

D2=P(H|¬E)−P(H)

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

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.

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From the old discussion page:

## Talk:Conservation of expected evidence

Regarding tilde versus overbar: I noticed that the Bayes’ theorem page uses \neg, resulting in an ¬ character, for that purpose. Should we use that here (including in the inline plain-text renderings) for consistency? (We should probably standardize on one such character to signify negation on all wiki pages, whether ~ or ¬.) —Adam Atlas 17:12, 25 August 2010 (UTC)

\neg it is, given that it seems to be more standard, and latex here renders tilde with inelegant amount of spacing around it. --Vladimir Nesov 20:41, 25 August 2010 (UTC)