I was reading Outlawing Anthropics and especially this subconversation has caught my attention. I got some ideas; but that thread is nearly four years old, so I am commenting here instead of there.
My version of the simplified situation: There is an intelligent rational agent (her name is Abby, she is well versed in Bayesian statistics) and there are two urns, each urn containing two marbles. Three of the marbles are green. They are macroscopic, so distinguishable, but not for Abby’s senses. Anyway, Abby can number them to be marbles 1, 2 and 3, she is just unable to “read” the number even on close examination. One marble is red, she can distinguish it, and it gets number 0. One urn has marbles 0 and 2, this is the “even” urn, the second has marbles 1 and 3, and is called odd. Again, Abby cannot distinguish urns without examining marbles. Now, assistent takes both urns to another room, computes 256th binary digit of exp(-1), and gets back with just one urn of corresponding parity. Abby is allowed to draw one marble (turns out it is green) and then urn is taken away and Abby is basically asked to state her subjective probability of the urn being odd (by accepting or refusing some bets). And only then she is told that in another room there is another person (Bart) who is being presented with the same choices after drawing the other marble from the very same urn. And finally, Abby is asked (informally) what is her averaged expectation of Bart’s subjective probability of the urn being odd (now that she sees her marble is green). And, if this average is different from her subjective probability, why is she not taking that value as an indirect evidence in her calculations (which clearly means that the assistent is just messing with her).
Assumptions are that neither Abby nor Bart have a clue about binary digits of exp(-1), they are not able to compute that far and so they assign prior probability of the urn being odd to 50%. Another assumption is that Abby and Bart both have chosen their marbles randomly, and in fact, they do not even know which one of them was drawing first. So there are 4 “possible” worlds, numbered by the marble Abby “would” have drawn, all of them appearing equally probable before the marble drawing.
Question is (of course) what subjective probability should Abby use when accepting/refusing bets. And to give a witty retort to assistant’s “why” question, where applicable; or else, to explain why Boltzmann brains are not that big obstacle to rationality.
And here I am, way over my time budget, having finished around one third of my planned comment. So I guess I shall leave you with questions for now, and I will resume commenting later.
Edit: Note to self: Do not forget to include http:// in links. RTFM.
Edit: “possible” worlds, numbered by marble Abby has drawn → “possible” worlds, numbered by marble Abby “would” have drawn
I was reading Outlawing Anthropics and especially this subconversation has caught my attention. I got some ideas; but that thread is nearly four years old, so I am commenting here instead of there.
My version of the simplified situation: There is an intelligent rational agent (her name is Abby, she is well versed in Bayesian statistics) and there are two urns, each urn containing two marbles. Three of the marbles are green. They are macroscopic, so distinguishable, but not for Abby’s senses. Anyway, Abby can number them to be marbles 1, 2 and 3, she is just unable to “read” the number even on close examination. One marble is red, she can distinguish it, and it gets number 0. One urn has marbles 0 and 2, this is the “even” urn, the second has marbles 1 and 3, and is called odd. Again, Abby cannot distinguish urns without examining marbles. Now, assistent takes both urns to another room, computes 256th binary digit of exp(-1), and gets back with just one urn of corresponding parity. Abby is allowed to draw one marble (turns out it is green) and then urn is taken away and Abby is basically asked to state her subjective probability of the urn being odd (by accepting or refusing some bets). And only then she is told that in another room there is another person (Bart) who is being presented with the same choices after drawing the other marble from the very same urn. And finally, Abby is asked (informally) what is her averaged expectation of Bart’s subjective probability of the urn being odd (now that she sees her marble is green). And, if this average is different from her subjective probability, why is she not taking that value as an indirect evidence in her calculations (which clearly means that the assistent is just messing with her).
Assumptions are that neither Abby nor Bart have a clue about binary digits of exp(-1), they are not able to compute that far and so they assign prior probability of the urn being odd to 50%. Another assumption is that Abby and Bart both have chosen their marbles randomly, and in fact, they do not even know which one of them was drawing first. So there are 4 “possible” worlds, numbered by the marble Abby “would” have drawn, all of them appearing equally probable before the marble drawing.
Question is (of course) what subjective probability should Abby use when accepting/refusing bets. And to give a witty retort to assistant’s “why” question, where applicable; or else, to explain why Boltzmann brains are not that big obstacle to rationality.
And here I am, way over my time budget, having finished around one third of my planned comment. So I guess I shall leave you with questions for now, and I will resume commenting later.
Edit: Note to self: Do not forget to include http:// in links. RTFM.
Edit: “possible” worlds, numbered by marble Abby has drawn → “possible” worlds, numbered by marble Abby “would” have drawn