# Difference between revisions of "Conservation of expected evidence"

m (removed duplication in a formula) |
MathieuRoy (talk | contribs) (→Blog posts: Mistakes with Conservation of Expected Evidence) |
||

Line 31: | Line 31: | ||

*[http://lesswrong.com/lw/ii/conservation_of_expected_evidence/ Conservation of Expected Evidence] | *[http://lesswrong.com/lw/ii/conservation_of_expected_evidence/ Conservation of Expected Evidence] | ||

+ | *[https://www.lesswrong.com/posts/zTfSXQracE7TW8x4w/mistakes-with-conservation-of-expected-evidence-1 Mistakes with Conservation of Expected Evidence] | ||

==See also== | ==See also== |

## Latest revision as of 08:59, 3 October 2019

**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:

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:

If the evidence is negative, the change is:

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

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.