Difference between revisions of "Conservation of expected evidence"
(marked as "featured") |
m (removed duplication in a formula) |
||
Line 8: | Line 8: | ||
<math>\begin{alignat}{2} | <math>\begin{alignat}{2} | ||
− | P(H) | + | P(H) & = P(H,E) + P(H,\neg{E}) \\ |
− | & = P(H,E) + P(H,\neg{E}) \\ | ||
& = P(H|E) \cdot P(E) + P(H|\neg{E}) \cdot P(\neg{E}) | & = P(H|E) \cdot P(E) + P(H|\neg{E}) \cdot P(\neg{E}) | ||
\end{alignat}</math> | \end{alignat}</math> |
Revision as of 00:51, 20 March 2012
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.