Difference between revisions of "Conservation of expected evidence"

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m (Yay, math works again! I like the tilde just for emphasis.)
m (changed formatting of negation from '~' to '¬')
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'''Conservation of expected evidence''' is a theorem that says: "for every expectation of evidence, there is an equal and opposite expectation of counterevidence".
 
'''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:
+
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:
  
<math>P(H|E) \cdot P(E) + P(H|\sim{E}) \cdot P(\sim{E})</math>
+
<math>P(H|E) \cdot P(E) + P(H|\neg{E}) \cdot P(\neg{E})</math>
  
 
But the prior probability of the hypothesis itself can be trivially broken up the same way:
 
But the prior probability of the hypothesis itself can be trivially broken up the same way:
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<math>\begin{alignat}{2}
 
<math>\begin{alignat}{2}
 
P(H) & = P(H) \\
 
P(H) & = P(H) \\
& = P(H,E) + P(H,\sim{E}) \\
+
& = P(H,E) + P(H,\neg{E}) \\
& = P(H|E) \cdot P(E) + P(H|\sim{E}) \cdot P(\sim{E})
+
& = P(H|E) \cdot P(E) + P(H|\neg{E}) \cdot P(\neg{E})
 
\end{alignat}</math>
 
\end{alignat}</math>
  
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If the evidence is negative, the change is:
 
If the evidence is negative, the change is:
  
<math>D_{2} = P(H|\sim{E})-P(H)\ </math>
+
<math>D_{2} = P(H|\neg{E})-P(H)\ </math>
  
 
Expectation of the change given positive evidence is equal to negated expectation of the change given counterevidence:
 
Expectation of the change given positive evidence is equal to negated expectation of the change given counterevidence:
  
<math>D_{1} \cdot P(E) = -D_{2} \cdot P(\sim{E})</math>
+
<math>D_{1} \cdot P(E) = -D_{2} \cdot P(\neg{E})</math>
  
 
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.  
 
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.  

Revision as of 07:44, 26 August 2010

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.

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See also