Conservation of Expected Evidence

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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 counterevidence. [1] 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.

From Conservation of Expected Evidence:

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.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.

These principlesConsider a hypothesis H and evidence (observation) E. Priorprobability 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:

P(H|E) ⋅ P(E) + P(HE) ⋅ PE)

But the prior probability of the hypothesis itself can be proven within standardtrivially 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 theory. From Absence of Evidence is Evidence of Absence:equal to the prior probability.

But in probability theory, absence of evidence is always evidence of absence. If E is a binary event and P(H | E) > P(H), i.e., seeing E increasesIn other way, if you expect the probability of H,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:

\(D_{2} = P(H|\neg{E})-P(H)\\)

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

D1 ⋅ P(E) =  − D2 ⋅ PE)

If you can anticipate in advance updating your belief in a particular direction, then P(H | ¬ E) < P(H), i.e., failure to observe E decreases theyou should just go ahead and update now. Once you know your destination, you are already there. On pain of paradox, a low probability of H . Theseeing strong evidence in one direction must be balanced by a high probability P(H) is a weighted mix of P(H | E) and P(H | ¬ E), and necessarily lies betweenobserving weak counterevidence in the two.other direction.

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    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 counterevidence. [1] The mere expectation of encountering evidence–before you've actually seen it–should not shift your prior beliefs.

    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 counterevidence. [1]

    From Conservation of Expected Evidence:

    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.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.

    A consequence of this principle is that absence of evidence is evidence of absence.

    These principles can be proven within standard probability theory. From Absence of Evidence is Evidence of Absence:

    But in probability theory, absence of evidence is always evidence of absence. If E is a binary event and P(H | E) > P(H), i.e., seeing E increases the probability of H, then P(H | ¬ E) < P(H), i.e., failure to observe E decreases the probability of H . The probability P(H) is a weighted mix of P(H | E) and P(H | ¬ E), and necessarily lies between the two.1

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