Bayes’ theorem updates on evidence by multiplying the old probability by the new evidence. The new evidence comes in the form of how likely the observation was under your hypothesis compared to the general rate.
So what comes out of bayes’ theorem here isn’t exactly an answer, but is an answer for each possible degree of “experience helped,” where you can then take the average to find out the likeliest helpfulness of experience.
P(guessed right) = 0.5. So if, for example, experience boosts you up to guessing right with P=0.6, then your prior probability rises by 0.6 / 0.5, while if you guessed wrong it would drop your prior probability by a factor of (1-0.6) / 0.5.
If you approximate experience as helping your probability of guessing right up to some value that’s uniformly distributed between 0.5 and 1, we (pretty obviously) get an average of 0.75. So guessing right multiplies P(experience helped) by a factor of approximately 3⁄2, while guessing wrong multiplies it by a factor of approximately 1⁄2.
What should your prior value of P(experience helped) be? That’s more complicated, since it deals with your memories and associated junk. It probably shouldn’t be more than 1⁄2, to be compatible with the guess I made a paragraph ago that P(guessed right | experience helped) can get close to 1 (that matters—check Bayes’ theorem), so if we do the same uniform approximation between 0 and 1⁄2 we get something like 0.25, going up to 0.375 if you guess right or down to 0.125 if you guess wrong.
P(guessed right) isn’t exactly 0.5 (most likely). We expand it as
P(experience helped (prior)) P(guessed right | experience helped) +
P(experience didn’t help (prior)) P(guessed right | experience didn’t help).
Of these, only P(guessed right | experience didn’t help) should be 0.5; P(guessed right | experience helped) should be higher. So on average P(guessed right) is somewhere in between depending on your priors.
So what comes out of bayes’ theorem here isn’t exactly an answer, but is an answer for each possible degree of “experience helped,” where you can then take the average to find out the likeliest helpfulness of experience.
P(experience helped | guessed right) = P(experience helped) * P(guessed right | experience helped) / P(guessed right)
Bayes’ theorem updates on evidence by multiplying the old probability by the new evidence. The new evidence comes in the form of how likely the observation was under your hypothesis compared to the general rate.
So what comes out of bayes’ theorem here isn’t exactly an answer, but is an answer for each possible degree of “experience helped,” where you can then take the average to find out the likeliest helpfulness of experience.
P(guessed right) = 0.5. So if, for example, experience boosts you up to guessing right with P=0.6, then your prior probability rises by 0.6 / 0.5, while if you guessed wrong it would drop your prior probability by a factor of (1-0.6) / 0.5.
If you approximate experience as helping your probability of guessing right up to some value that’s uniformly distributed between 0.5 and 1, we (pretty obviously) get an average of 0.75. So guessing right multiplies P(experience helped) by a factor of approximately 3⁄2, while guessing wrong multiplies it by a factor of approximately 1⁄2.
What should your prior value of P(experience helped) be? That’s more complicated, since it deals with your memories and associated junk. It probably shouldn’t be more than 1⁄2, to be compatible with the guess I made a paragraph ago that P(guessed right | experience helped) can get close to 1 (that matters—check Bayes’ theorem), so if we do the same uniform approximation between 0 and 1⁄2 we get something like 0.25, going up to 0.375 if you guess right or down to 0.125 if you guess wrong.
EDIT: not quite right, see Misha’s comment.
P(guessed right) isn’t exactly 0.5 (most likely). We expand it as
P(experience helped (prior)) P(guessed right | experience helped) + P(experience didn’t help (prior)) P(guessed right | experience didn’t help).
Of these, only P(guessed right | experience didn’t help) should be 0.5; P(guessed right | experience helped) should be higher. So on average P(guessed right) is somewhere in between depending on your priors.
Right, thanks. Pushing the uniform approximations to the point of silliness I get 4⁄7 for a rough value (with the prior then being 2⁄7).
This was the paragraph I needed, thanks.