Conservation of Expected Evidence
Followup to: Absence of Evidence Is Evidence of Absence.
Friedrich Spee von Langenfeld, a priest who heard the confessions of condemned witches, wrote in 1631 the Cautio Criminalis ('prudence in criminal cases') in which he bitingly described the decision tree for condemning accused witches: If the witch had led an evil and improper life, she was guilty; if she had led a good and proper life, this too was a proof, for witches dissemble and try to appear especially virtuous. After the woman was put in prison: if she was afraid, this proved her guilt; if she was not afraid, this proved her guilt, for witches characteristically pretend innocence and wear a bold front. Or on hearing of a denunciation of witchcraft against her, she might seek flight or remain; if she ran, that proved her guilt; if she remained, the devil had detained her so she could not get away.
Spee acted as confessor to many witches; he was thus in a position to observe every branch of the accusation tree, that no matter what the accused witch said or did, it was held a proof against her. In any individual case, you would only hear one branch of the dilemma. It is for this reason that scientists write down their experimental predictions in advance.
But you can't have it both ways—as a matter of probability theory, not mere fairness. The rule that "absence of evidence is evidence of absence" is a special case of a more general law, which I would name Conservation of Expected Evidence: The expectation of the posterior probability, after viewing the evidence, must equal the prior probability.
P(H) = P(H)
P(H) = P(H,E) + P(H,~E)
P(H) = P(H|E)*P(E) + P(H|~E)*P(~E)
Therefore, for every expectation of evidence, there is an equal and opposite expectation of counterevidence.
If you expect a strong probability of seeing weak evidence in one direction, it must be balanced by a weak expectation of seeing strong evidence in the other direction. If you're very confident in your theory, and therefore anticipate seeing an outcome that matches your hypothesis, this can only provide a very small increment to your belief (it is already close to 1); but the unexpected failure of your prediction would (and must) deal your confidence a huge blow. On average, you must expect to be exactly as confident as when you started out. Equivalently, the mere expectation of encountering evidence—before you've actually seen it—should not shift your prior beliefs. (Again, if this is not intuitively obvious, see An Intuitive Explanation of Bayesian Reasoning.)
So if you claim that "no sabotage" is evidence for the existence of a Japanese-American Fifth Column, you must conversely hold that seeing sabotage would argue against a Fifth Column. If you claim that "a good and proper life" is evidence that a woman is a witch, then an evil and improper life must be evidence that she is not a witch. If you argue that God, to test humanity's faith, refuses to reveal His existence, then the miracles described in the Bible must argue against the existence of God.
Doesn't quite sound right, does it? Pay attention to that feeling of this seems a little forced, that quiet strain in the back of your mind. It's important.
For a true Bayesian, it is impossible to seek evidence that confirms a theory. There is no possible plan you can devise, no clever strategy, no cunning device, by which you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before. You can only ever seek evidence to test a theory, not to confirm it.
This realization can take quite a load off your mind. You need not worry about how to interpret every possible experimental result to confirm your theory. You needn't bother planning how to make any given iota of evidence confirm your theory, because you know that for every expectation of evidence, there is an equal and oppositive expectation of counterevidence. If you try to weaken the counterevidence of a possible "abnormal" observation, you can only do it by weakening the support of a "normal" observation, to a precisely equal and opposite degree. It is a zero-sum game. No matter how you connive, no matter how you argue, no matter how you strategize, you can't possibly expect the resulting game plan to shift your beliefs (on average) in a particular direction.
You might as well sit back and relax while you wait for the evidence to come in.
...human psychology is so screwed up.
Part of the sequence Mysterious Answers to Mysterious Questions
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